![Page 1: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/1.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
The Natural Logarithm and ExponentialFunctions
Bander Almutairi
King Saud University
26 Sept 2013
![Page 2: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/2.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
1 General Exponential Function
2 Derivatives of General Exponential Function
3 Integration of General Exponential Functions
4 General Logarithm Functions
5 Derivative of General Logarithmic Function
![Page 3: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/3.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Definition
If a is positive, then a function f (x) = ax is called theexponential function with base a and x called the exponent.
The function f (x) is also called general exponential function.Examples: f (x) = 2x , g(x) = 6x .* If the exponent is a rational number r , then
ax = e ln(ar ) = er ln(a), a > 0.
* Relation between general and natural exponential is
ax = ex ln(a), a > 0, x ∈ R.
![Page 4: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/4.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Definition
If a is positive, then a function f (x) = ax is called theexponential function with base a and x called the exponent.
The function f (x) is also called general exponential function.
Examples: f (x) = 2x , g(x) = 6x .* If the exponent is a rational number r , then
ax = e ln(ar ) = er ln(a), a > 0.
* Relation between general and natural exponential is
ax = ex ln(a), a > 0, x ∈ R.
![Page 5: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/5.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Definition
If a is positive, then a function f (x) = ax is called theexponential function with base a and x called the exponent.
The function f (x) is also called general exponential function.Examples: f (x) = 2x , g(x) = 6x .
* If the exponent is a rational number r , then
ax = e ln(ar ) = er ln(a), a > 0.
* Relation between general and natural exponential is
ax = ex ln(a), a > 0, x ∈ R.
![Page 6: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/6.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Definition
If a is positive, then a function f (x) = ax is called theexponential function with base a and x called the exponent.
The function f (x) is also called general exponential function.Examples: f (x) = 2x , g(x) = 6x .* If the exponent is a rational number r , then
ax = e ln(ar ) = er ln(a), a > 0.
* Relation between general and natural exponential is
ax = ex ln(a), a > 0, x ∈ R.
![Page 7: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/7.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Definition
If a is positive, then a function f (x) = ax is called theexponential function with base a and x called the exponent.
The function f (x) is also called general exponential function.Examples: f (x) = 2x , g(x) = 6x .* If the exponent is a rational number r , then
ax = e ln(ar ) = er ln(a), a > 0.
* Relation between general and natural exponential is
ax = ex ln(a), a > 0, x ∈ R.
![Page 8: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/8.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Definition
If a is positive, then a function f (x) = ax is called theexponential function with base a and x called the exponent.
The function f (x) is also called general exponential function.Examples: f (x) = 2x , g(x) = 6x .* If the exponent is a rational number r , then
ax = e ln(ar ) = er ln(a), a > 0.
* Relation between general and natural exponential is
ax = ex ln(a), a > 0, x ∈ R.
![Page 9: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/9.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Definition
If a is positive, then a function f (x) = ax is called theexponential function with base a and x called the exponent.
The function f (x) is also called general exponential function.Examples: f (x) = 2x , g(x) = 6x .* If the exponent is a rational number r , then
ax = e ln(ar ) = er ln(a), a > 0.
* Relation between general and natural exponential is
ax = ex ln(a), a > 0, x ∈ R.
![Page 10: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/10.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Laws of Exponents:
Let a > 0 and b > 0. If u and v are anyreal numbers
(i) auav = au+v .
(ii) au/av = au−v .
(iii) (au)v = auv .
(iv) (ab)u = aubu.
(v) (a/b)u = au/bu.
![Page 11: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/11.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Laws of Exponents: Let a > 0 and b > 0. If u and v are anyreal numbers
(i) auav = au+v .
(ii) au/av = au−v .
(iii) (au)v = auv .
(iv) (ab)u = aubu.
(v) (a/b)u = au/bu.
![Page 12: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/12.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Laws of Exponents: Let a > 0 and b > 0. If u and v are anyreal numbers
(i) auav = au+v .
(ii) au/av = au−v .
(iii) (au)v = auv .
(iv) (ab)u = aubu.
(v) (a/b)u = au/bu.
![Page 13: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/13.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Laws of Exponents: Let a > 0 and b > 0. If u and v are anyreal numbers
(i) auav = au+v .
(ii) au/av = au−v .
(iii) (au)v = auv .
(iv) (ab)u = aubu.
(v) (a/b)u = au/bu.
![Page 14: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/14.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Laws of Exponents: Let a > 0 and b > 0. If u and v are anyreal numbers
(i) auav = au+v .
(ii) au/av = au−v .
(iii) (au)v = auv .
(iv) (ab)u = aubu.
(v) (a/b)u = au/bu.
![Page 15: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/15.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Laws of Exponents: Let a > 0 and b > 0. If u and v are anyreal numbers
(i) auav = au+v .
(ii) au/av = au−v .
(iii) (au)v = auv .
(iv) (ab)u = aubu.
(v) (a/b)u = au/bu.
![Page 16: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/16.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Laws of Exponents: Let a > 0 and b > 0. If u and v are anyreal numbers
(i) auav = au+v .
(ii) au/av = au−v .
(iii) (au)v = auv .
(iv) (ab)u = aubu.
(v) (a/b)u = au/bu.
![Page 17: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/17.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, then
d
dx(ax) = ax . ln(a).
Proof:
d
dx(ax) =
d
dx
(ex ln(a)
)= ex ln(a).
d
dx(x ln(a))
= ex ln(a). ln(a) = ax . ln(a)
Examples:
(a) ddx (5x) = 5x ln(5).
(b) ddx
(3√2x)
= 3√2x ln(3) d
dx (√
2x) = 3√2x ln(3)
(1
2√x
).
![Page 18: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/18.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, thend
dx(ax) = ax . ln(a).
Proof:
d
dx(ax) =
d
dx
(ex ln(a)
)= ex ln(a).
d
dx(x ln(a))
= ex ln(a). ln(a) = ax . ln(a)
Examples:
(a) ddx (5x) = 5x ln(5).
(b) ddx
(3√2x)
= 3√2x ln(3) d
dx (√
2x) = 3√2x ln(3)
(1
2√x
).
![Page 19: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/19.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, thend
dx(ax) = ax . ln(a).
Proof:
d
dx(ax) =
d
dx
(ex ln(a)
)= ex ln(a).
d
dx(x ln(a))
= ex ln(a). ln(a) = ax . ln(a)
Examples:
(a) ddx (5x) = 5x ln(5).
(b) ddx
(3√2x)
= 3√2x ln(3) d
dx (√
2x) = 3√2x ln(3)
(1
2√x
).
![Page 20: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/20.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, thend
dx(ax) = ax . ln(a).
Proof:
d
dx(ax)
=d
dx
(ex ln(a)
)= ex ln(a).
d
dx(x ln(a))
= ex ln(a). ln(a) = ax . ln(a)
Examples:
(a) ddx (5x) = 5x ln(5).
(b) ddx
(3√2x)
= 3√2x ln(3) d
dx (√
2x) = 3√2x ln(3)
(1
2√x
).
![Page 21: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/21.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, thend
dx(ax) = ax . ln(a).
Proof:
d
dx(ax) =
d
dx
(ex ln(a)
)
= ex ln(a).d
dx(x ln(a))
= ex ln(a). ln(a) = ax . ln(a)
Examples:
(a) ddx (5x) = 5x ln(5).
(b) ddx
(3√2x)
= 3√2x ln(3) d
dx (√
2x) = 3√2x ln(3)
(1
2√x
).
![Page 22: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/22.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, thend
dx(ax) = ax . ln(a).
Proof:
d
dx(ax) =
d
dx
(ex ln(a)
)= ex ln(a).
d
dx(x ln(a))
= ex ln(a). ln(a) = ax . ln(a)
Examples:
(a) ddx (5x) = 5x ln(5).
(b) ddx
(3√2x)
= 3√2x ln(3) d
dx (√
2x) = 3√2x ln(3)
(1
2√x
).
![Page 23: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/23.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, thend
dx(ax) = ax . ln(a).
Proof:
d
dx(ax) =
d
dx
(ex ln(a)
)= ex ln(a).
d
dx(x ln(a))
= ex ln(a). ln(a)
= ax . ln(a)
Examples:
(a) ddx (5x) = 5x ln(5).
(b) ddx
(3√2x)
= 3√2x ln(3) d
dx (√
2x) = 3√2x ln(3)
(1
2√x
).
![Page 24: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/24.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, thend
dx(ax) = ax . ln(a).
Proof:
d
dx(ax) =
d
dx
(ex ln(a)
)= ex ln(a).
d
dx(x ln(a))
= ex ln(a). ln(a) = ax . ln(a)
Examples:
(a) ddx (5x) = 5x ln(5).
(b) ddx
(3√2x)
= 3√2x ln(3) d
dx (√
2x) = 3√2x ln(3)
(1
2√x
).
![Page 25: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/25.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, thend
dx(ax) = ax . ln(a).
Proof:
d
dx(ax) =
d
dx
(ex ln(a)
)= ex ln(a).
d
dx(x ln(a))
= ex ln(a). ln(a) = ax . ln(a)
Examples:
(a) ddx (5x) = 5x ln(5).
(b) ddx
(3√2x)
= 3√2x ln(3) d
dx (√
2x) = 3√2x ln(3)
(1
2√x
).
![Page 26: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/26.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, thend
dx(ax) = ax . ln(a).
Proof:
d
dx(ax) =
d
dx
(ex ln(a)
)= ex ln(a).
d
dx(x ln(a))
= ex ln(a). ln(a) = ax . ln(a)
Examples:
(a) ddx (5x) = 5x ln(5).
(b) ddx
(3√2x)
= 3√2x ln(3) d
dx (√
2x) = 3√2x ln(3)
(1
2√x
).
![Page 27: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/27.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, thend
dx(ax) = ax . ln(a).
Proof:
d
dx(ax) =
d
dx
(ex ln(a)
)= ex ln(a).
d
dx(x ln(a))
= ex ln(a). ln(a) = ax . ln(a)
Examples:
(a) ddx (5x) = 5x ln(5).
(b) ddx
(3√2x)
= 3√2x ln(3) d
dx (√
2x) = 3√2x ln(3)
(1
2√x
).
![Page 28: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/28.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, thend
dx(ax) = ax . ln(a).
Proof:
d
dx(ax) =
d
dx
(ex ln(a)
)= ex ln(a).
d
dx(x ln(a))
= ex ln(a). ln(a) = ax . ln(a)
Examples:
(a) ddx (5x) = 5x ln(5).
(b) ddx
(3√2x)
= 3√2x ln(3) d
dx (√
2x)
= 3√2x ln(3)
(1
2√x
).
![Page 29: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/29.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, thend
dx(ax) = ax . ln(a).
Proof:
d
dx(ax) =
d
dx
(ex ln(a)
)= ex ln(a).
d
dx(x ln(a))
= ex ln(a). ln(a) = ax . ln(a)
Examples:
(a) ddx (5x) = 5x ln(5).
(b) ddx
(3√2x)
= 3√2x ln(3) d
dx (√
2x) = 3√2x ln(3)
(1
2√x
).
![Page 30: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/30.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
* If a > 1,
then ln(a) > 0,therefore,
d
dx(ax) = ax ln(a) > 0.
Hence f (x) = ax is increasing function on the interval (1,∞).
* If 0 < a < 1, then ln(a) < 0, therefore,
d
dx(ax) = ax ln(a) < 0.
Hence f (x) = ax is decreasing function on the interval (0, 1)
![Page 31: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/31.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
* If a > 1, then ln(a) > 0,
therefore,
d
dx(ax) = ax ln(a) > 0.
Hence f (x) = ax is increasing function on the interval (1,∞).
* If 0 < a < 1, then ln(a) < 0, therefore,
d
dx(ax) = ax ln(a) < 0.
Hence f (x) = ax is decreasing function on the interval (0, 1)
![Page 32: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/32.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
* If a > 1, then ln(a) > 0,therefore,
d
dx(ax) = ax ln(a) > 0.
Hence f (x) = ax is increasing function on the interval (1,∞).
* If 0 < a < 1, then ln(a) < 0, therefore,
d
dx(ax) = ax ln(a) < 0.
Hence f (x) = ax is decreasing function on the interval (0, 1)
![Page 33: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/33.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
* If a > 1, then ln(a) > 0,therefore,
d
dx(ax) = ax ln(a) > 0.
Hence f (x) = ax is increasing function on the interval (1,∞).
* If 0 < a < 1, then ln(a) < 0, therefore,
d
dx(ax) = ax ln(a) < 0.
Hence f (x) = ax is decreasing function on the interval (0, 1)
![Page 34: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/34.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
* If a > 1, then ln(a) > 0,therefore,
d
dx(ax) = ax ln(a) > 0.
Hence f (x) = ax is increasing function on the interval (1,∞).
* If 0 < a < 1, then ln(a) < 0, therefore,
d
dx(ax) = ax ln(a) < 0.
Hence f (x) = ax is decreasing function on the interval (0, 1)
![Page 35: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/35.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
* If a > 1, then ln(a) > 0,therefore,
d
dx(ax) = ax ln(a) > 0.
Hence f (x) = ax is increasing function on the interval (1,∞).
* If 0 < a < 1,
then ln(a) < 0, therefore,
d
dx(ax) = ax ln(a) < 0.
Hence f (x) = ax is decreasing function on the interval (0, 1)
![Page 36: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/36.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
* If a > 1, then ln(a) > 0,therefore,
d
dx(ax) = ax ln(a) > 0.
Hence f (x) = ax is increasing function on the interval (1,∞).
* If 0 < a < 1, then ln(a) < 0,
therefore,
d
dx(ax) = ax ln(a) < 0.
Hence f (x) = ax is decreasing function on the interval (0, 1)
![Page 37: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/37.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
* If a > 1, then ln(a) > 0,therefore,
d
dx(ax) = ax ln(a) > 0.
Hence f (x) = ax is increasing function on the interval (1,∞).
* If 0 < a < 1, then ln(a) < 0, therefore,
d
dx(ax) = ax ln(a) < 0.
Hence f (x) = ax is decreasing function on the interval (0, 1)
![Page 38: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/38.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
* If a > 1, then ln(a) > 0,therefore,
d
dx(ax) = ax ln(a) > 0.
Hence f (x) = ax is increasing function on the interval (1,∞).
* If 0 < a < 1, then ln(a) < 0, therefore,
d
dx(ax) = ax ln(a) < 0.
Hence f (x) = ax is decreasing function on the interval (0, 1)
![Page 39: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/39.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
* If a > 1, then ln(a) > 0,therefore,
d
dx(ax) = ax ln(a) > 0.
Hence f (x) = ax is increasing function on the interval (1,∞).
* If 0 < a < 1, then ln(a) < 0, therefore,
d
dx(ax) = ax ln(a) < 0.
Hence f (x) = ax is decreasing function on the interval (0, 1)
![Page 40: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/40.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Find f ′(x) in each of the following:
(a) f (x) = 5√x3 .
f ′(x) = 5√x3 ln(5)
d
dx
(√x3)
= 5√x3 ln(5).
3x2
2√x3.
(b) f (x) = 71−2x5.
f ′(x) = 71−2x5
ln(7)d
dx
(1− 2x5
)= 71−2x
5ln(7)(−10x4).
![Page 41: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/41.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Find f ′(x) in each of the following:
(a) f (x) = 5√x3 .
f ′(x) = 5√x3 ln(5)
d
dx
(√x3)
= 5√x3 ln(5).
3x2
2√x3.
(b) f (x) = 71−2x5.
f ′(x) = 71−2x5
ln(7)d
dx
(1− 2x5
)= 71−2x
5ln(7)(−10x4).
![Page 42: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/42.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Find f ′(x) in each of the following:
(a) f (x) = 5√x3 .
f ′(x)
= 5√x3 ln(5)
d
dx
(√x3)
= 5√x3 ln(5).
3x2
2√x3.
(b) f (x) = 71−2x5.
f ′(x) = 71−2x5
ln(7)d
dx
(1− 2x5
)= 71−2x
5ln(7)(−10x4).
![Page 43: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/43.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Find f ′(x) in each of the following:
(a) f (x) = 5√x3 .
f ′(x) = 5√x3 ln(5)
d
dx
(√x3)
= 5√x3 ln(5).
3x2
2√x3.
(b) f (x) = 71−2x5.
f ′(x) = 71−2x5
ln(7)d
dx
(1− 2x5
)= 71−2x
5ln(7)(−10x4).
![Page 44: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/44.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Find f ′(x) in each of the following:
(a) f (x) = 5√x3 .
f ′(x) = 5√x3 ln(5)
d
dx
(√x3)
= 5√x3 ln(5).
3x2
2√x3.
(b) f (x) = 71−2x5.
f ′(x) = 71−2x5
ln(7)d
dx
(1− 2x5
)= 71−2x
5ln(7)(−10x4).
![Page 45: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/45.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Find f ′(x) in each of the following:
(a) f (x) = 5√x3 .
f ′(x) = 5√x3 ln(5)
d
dx
(√x3)
= 5√x3 ln(5).
3x2
2√x3.
(b) f (x) = 71−2x5.
f ′(x) = 71−2x5
ln(7)d
dx
(1− 2x5
)= 71−2x
5ln(7)(−10x4).
![Page 46: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/46.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Find f ′(x) in each of the following:
(a) f (x) = 5√x3 .
f ′(x) = 5√x3 ln(5)
d
dx
(√x3)
= 5√x3 ln(5).
3x2
2√x3.
(b) f (x) = 71−2x5.
f ′(x)
= 71−2x5
ln(7)d
dx
(1− 2x5
)= 71−2x
5ln(7)(−10x4).
![Page 47: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/47.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Find f ′(x) in each of the following:
(a) f (x) = 5√x3 .
f ′(x) = 5√x3 ln(5)
d
dx
(√x3)
= 5√x3 ln(5).
3x2
2√x3.
(b) f (x) = 71−2x5.
f ′(x) = 71−2x5
ln(7)d
dx
(1− 2x5
)
= 71−2x5
ln(7)(−10x4).
![Page 48: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/48.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Find f ′(x) in each of the following:
(a) f (x) = 5√x3 .
f ′(x) = 5√x3 ln(5)
d
dx
(√x3)
= 5√x3 ln(5).
3x2
2√x3.
(b) f (x) = 71−2x5.
f ′(x) = 71−2x5
ln(7)d
dx
(1− 2x5
)= 71−2x
5ln(7)(−10x4).
![Page 49: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/49.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
(c) f (x) = (2x + 2−x)9.
f ′(x) = 9(2x + 2−x)8.d
dx(2x + 2−x)
= 9(2x + 2−x)8.
[d
dx(2x) +
d
dx(2−x)
]= 9(2x + 2−x)8
[2x ln(2)
d
dx(x) + 2−x ln(2)
d
dx(−x)
]= 9(2x + 2−x)8
[2x ln(2)− 2−x ln(2)
].
![Page 50: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/50.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
(c) f (x) = (2x + 2−x)9.
f ′(x)
= 9(2x + 2−x)8.d
dx(2x + 2−x)
= 9(2x + 2−x)8.
[d
dx(2x) +
d
dx(2−x)
]= 9(2x + 2−x)8
[2x ln(2)
d
dx(x) + 2−x ln(2)
d
dx(−x)
]= 9(2x + 2−x)8
[2x ln(2)− 2−x ln(2)
].
![Page 51: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/51.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
(c) f (x) = (2x + 2−x)9.
f ′(x) = 9(2x + 2−x)8.d
dx(2x + 2−x)
= 9(2x + 2−x)8.
[d
dx(2x) +
d
dx(2−x)
]= 9(2x + 2−x)8
[2x ln(2)
d
dx(x) + 2−x ln(2)
d
dx(−x)
]= 9(2x + 2−x)8
[2x ln(2)− 2−x ln(2)
].
![Page 52: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/52.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
(c) f (x) = (2x + 2−x)9.
f ′(x) = 9(2x + 2−x)8.d
dx(2x + 2−x)
= 9(2x + 2−x)8.
[d
dx(2x) +
d
dx(2−x)
]
= 9(2x + 2−x)8[
2x ln(2)d
dx(x) + 2−x ln(2)
d
dx(−x)
]= 9(2x + 2−x)8
[2x ln(2)− 2−x ln(2)
].
![Page 53: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/53.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
(c) f (x) = (2x + 2−x)9.
f ′(x) = 9(2x + 2−x)8.d
dx(2x + 2−x)
= 9(2x + 2−x)8.
[d
dx(2x) +
d
dx(2−x)
]= 9(2x + 2−x)8
[2x ln(2)
d
dx(x) + 2−x ln(2)
d
dx(−x)
]
= 9(2x + 2−x)8[2x ln(2)− 2−x ln(2)
].
![Page 54: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/54.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
(c) f (x) = (2x + 2−x)9.
f ′(x) = 9(2x + 2−x)8.d
dx(2x + 2−x)
= 9(2x + 2−x)8.
[d
dx(2x) +
d
dx(2−x)
]= 9(2x + 2−x)8
[2x ln(2)
d
dx(x) + 2−x ln(2)
d
dx(−x)
]= 9(2x + 2−x)8
[2x ln(2)− 2−x ln(2)
].
![Page 55: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/55.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, then
∫audu =
1
ln(a).au + c .
Proof:We know that d
du (au) = au ln(a). Integrating both sides∫d
du(au)du =
∫au ln(a)du,
so we get
au + c = ln(a)
∫audu.
Hence ∫audu =
1
ln(a).au + c .
![Page 56: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/56.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, then ∫audu =
1
ln(a).au + c .
Proof:We know that d
du (au) = au ln(a). Integrating both sides∫d
du(au)du =
∫au ln(a)du,
so we get
au + c = ln(a)
∫audu.
Hence ∫audu =
1
ln(a).au + c .
![Page 57: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/57.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, then ∫audu =
1
ln(a).au + c .
Proof:
We know that ddu (au) = au ln(a). Integrating both sides∫
d
du(au)du =
∫au ln(a)du,
so we get
au + c = ln(a)
∫audu.
Hence ∫audu =
1
ln(a).au + c .
![Page 58: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/58.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, then ∫audu =
1
ln(a).au + c .
Proof:We know that d
du (au)
= au ln(a). Integrating both sides∫d
du(au)du =
∫au ln(a)du,
so we get
au + c = ln(a)
∫audu.
Hence ∫audu =
1
ln(a).au + c .
![Page 59: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/59.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, then ∫audu =
1
ln(a).au + c .
Proof:We know that d
du (au) = au ln(a).
Integrating both sides∫d
du(au)du =
∫au ln(a)du,
so we get
au + c = ln(a)
∫audu.
Hence ∫audu =
1
ln(a).au + c .
![Page 60: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/60.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, then ∫audu =
1
ln(a).au + c .
Proof:We know that d
du (au) = au ln(a). Integrating both sides
∫d
du(au)du =
∫au ln(a)du,
so we get
au + c = ln(a)
∫audu.
Hence ∫audu =
1
ln(a).au + c .
![Page 61: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/61.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, then ∫audu =
1
ln(a).au + c .
Proof:We know that d
du (au) = au ln(a). Integrating both sides∫d
du(au)du
=
∫au ln(a)du,
so we get
au + c = ln(a)
∫audu.
Hence ∫audu =
1
ln(a).au + c .
![Page 62: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/62.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, then ∫audu =
1
ln(a).au + c .
Proof:We know that d
du (au) = au ln(a). Integrating both sides∫d
du(au)du =
∫au ln(a)du,
so we get
au + c = ln(a)
∫audu.
Hence ∫audu =
1
ln(a).au + c .
![Page 63: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/63.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, then ∫audu =
1
ln(a).au + c .
Proof:We know that d
du (au) = au ln(a). Integrating both sides∫d
du(au)du =
∫au ln(a)du,
so we get
au + c = ln(a)
∫audu.
Hence ∫audu =
1
ln(a).au + c .
![Page 64: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/64.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, then ∫audu =
1
ln(a).au + c .
Proof:We know that d
du (au) = au ln(a). Integrating both sides∫d
du(au)du =
∫au ln(a)du,
so we get
au + c = ln(a)
∫audu.
Hence ∫audu =
1
ln(a).au + c .
![Page 65: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/65.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, then ∫audu =
1
ln(a).au + c .
Proof:We know that d
du (au) = au ln(a). Integrating both sides∫d
du(au)du =
∫au ln(a)du,
so we get
au + c = ln(a)
∫audu.
Hence
∫audu =
1
ln(a).au + c .
![Page 66: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/66.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Theorem
If a > 0, then ∫audu =
1
ln(a).au + c .
Proof:We know that d
du (au) = au ln(a). Integrating both sides∫d
du(au)du =
∫au ln(a)du,
so we get
au + c = ln(a)
∫audu.
Hence ∫audu =
1
ln(a).au + c .
![Page 67: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/67.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples:
Evaluate the following integrals.[1]∫ 0
−35xdx =
[1
ln(5)5x]0−3
=1
ln(5)
[1− 5−3
]=
124
125 ln(5).
[2] ∫5tan(x)
cos2(x)dx .
Put = tan(x), so du = sec2(x)dx = 1cos2(x)
dx . Hence,
∫5tan(x)
cos2(x)dx =
∫5udu =
1
ln(5)5u + c =
1
ln(5)5tan(x) + c.
![Page 68: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/68.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Evaluate the following integrals.
[1]∫ 0
−35xdx =
[1
ln(5)5x]0−3
=1
ln(5)
[1− 5−3
]=
124
125 ln(5).
[2] ∫5tan(x)
cos2(x)dx .
Put = tan(x), so du = sec2(x)dx = 1cos2(x)
dx . Hence,
∫5tan(x)
cos2(x)dx =
∫5udu =
1
ln(5)5u + c =
1
ln(5)5tan(x) + c.
![Page 69: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/69.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Evaluate the following integrals.[1]∫ 0
−35xdx
=
[1
ln(5)5x]0−3
=1
ln(5)
[1− 5−3
]=
124
125 ln(5).
[2] ∫5tan(x)
cos2(x)dx .
Put = tan(x), so du = sec2(x)dx = 1cos2(x)
dx . Hence,
∫5tan(x)
cos2(x)dx =
∫5udu =
1
ln(5)5u + c =
1
ln(5)5tan(x) + c.
![Page 70: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/70.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Evaluate the following integrals.[1]∫ 0
−35xdx =
[1
ln(5)5x]0−3
=1
ln(5)
[1− 5−3
]=
124
125 ln(5).
[2] ∫5tan(x)
cos2(x)dx .
Put = tan(x), so du = sec2(x)dx = 1cos2(x)
dx . Hence,
∫5tan(x)
cos2(x)dx =
∫5udu =
1
ln(5)5u + c =
1
ln(5)5tan(x) + c.
![Page 71: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/71.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Evaluate the following integrals.[1]∫ 0
−35xdx =
[1
ln(5)5x]0−3
=1
ln(5)
[1− 5−3
]
=124
125 ln(5).
[2] ∫5tan(x)
cos2(x)dx .
Put = tan(x), so du = sec2(x)dx = 1cos2(x)
dx . Hence,
∫5tan(x)
cos2(x)dx =
∫5udu =
1
ln(5)5u + c =
1
ln(5)5tan(x) + c.
![Page 72: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/72.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Evaluate the following integrals.[1]∫ 0
−35xdx =
[1
ln(5)5x]0−3
=1
ln(5)
[1− 5−3
]=
124
125 ln(5).
[2] ∫5tan(x)
cos2(x)dx .
Put = tan(x), so du = sec2(x)dx = 1cos2(x)
dx . Hence,
∫5tan(x)
cos2(x)dx =
∫5udu =
1
ln(5)5u + c =
1
ln(5)5tan(x) + c.
![Page 73: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/73.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Evaluate the following integrals.[1]∫ 0
−35xdx =
[1
ln(5)5x]0−3
=1
ln(5)
[1− 5−3
]=
124
125 ln(5).
[2] ∫5tan(x)
cos2(x)dx .
Put = tan(x), so du = sec2(x)dx = 1cos2(x)
dx . Hence,
∫5tan(x)
cos2(x)dx =
∫5udu =
1
ln(5)5u + c =
1
ln(5)5tan(x) + c.
![Page 74: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/74.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Evaluate the following integrals.[1]∫ 0
−35xdx =
[1
ln(5)5x]0−3
=1
ln(5)
[1− 5−3
]=
124
125 ln(5).
[2] ∫5tan(x)
cos2(x)dx .
Put = tan(x),
so du = sec2(x)dx = 1cos2(x)
dx . Hence,
∫5tan(x)
cos2(x)dx =
∫5udu =
1
ln(5)5u + c =
1
ln(5)5tan(x) + c.
![Page 75: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/75.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Evaluate the following integrals.[1]∫ 0
−35xdx =
[1
ln(5)5x]0−3
=1
ln(5)
[1− 5−3
]=
124
125 ln(5).
[2] ∫5tan(x)
cos2(x)dx .
Put = tan(x), so du
= sec2(x)dx = 1cos2(x)
dx . Hence,
∫5tan(x)
cos2(x)dx =
∫5udu =
1
ln(5)5u + c =
1
ln(5)5tan(x) + c.
![Page 76: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/76.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Evaluate the following integrals.[1]∫ 0
−35xdx =
[1
ln(5)5x]0−3
=1
ln(5)
[1− 5−3
]=
124
125 ln(5).
[2] ∫5tan(x)
cos2(x)dx .
Put = tan(x), so du = sec2(x)dx
= 1cos2(x)
dx . Hence,
∫5tan(x)
cos2(x)dx =
∫5udu =
1
ln(5)5u + c =
1
ln(5)5tan(x) + c.
![Page 77: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/77.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Evaluate the following integrals.[1]∫ 0
−35xdx =
[1
ln(5)5x]0−3
=1
ln(5)
[1− 5−3
]=
124
125 ln(5).
[2] ∫5tan(x)
cos2(x)dx .
Put = tan(x), so du = sec2(x)dx = 1cos2(x)
dx .
Hence,
∫5tan(x)
cos2(x)dx =
∫5udu =
1
ln(5)5u + c =
1
ln(5)5tan(x) + c.
![Page 78: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/78.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Evaluate the following integrals.[1]∫ 0
−35xdx =
[1
ln(5)5x]0−3
=1
ln(5)
[1− 5−3
]=
124
125 ln(5).
[2] ∫5tan(x)
cos2(x)dx .
Put = tan(x), so du = sec2(x)dx = 1cos2(x)
dx . Hence,
∫5tan(x)
cos2(x)dx =
∫5udu =
1
ln(5)5u + c =
1
ln(5)5tan(x) + c.
![Page 79: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/79.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Evaluate the following integrals.[1]∫ 0
−35xdx =
[1
ln(5)5x]0−3
=1
ln(5)
[1− 5−3
]=
124
125 ln(5).
[2] ∫5tan(x)
cos2(x)dx .
Put = tan(x), so du = sec2(x)dx = 1cos2(x)
dx . Hence,
∫5tan(x)
cos2(x)dx
=
∫5udu =
1
ln(5)5u + c =
1
ln(5)5tan(x) + c.
![Page 80: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/80.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Evaluate the following integrals.[1]∫ 0
−35xdx =
[1
ln(5)5x]0−3
=1
ln(5)
[1− 5−3
]=
124
125 ln(5).
[2] ∫5tan(x)
cos2(x)dx .
Put = tan(x), so du = sec2(x)dx = 1cos2(x)
dx . Hence,
∫5tan(x)
cos2(x)dx =
∫5udu
=1
ln(5)5u + c =
1
ln(5)5tan(x) + c.
![Page 81: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/81.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Evaluate the following integrals.[1]∫ 0
−35xdx =
[1
ln(5)5x]0−3
=1
ln(5)
[1− 5−3
]=
124
125 ln(5).
[2] ∫5tan(x)
cos2(x)dx .
Put = tan(x), so du = sec2(x)dx = 1cos2(x)
dx . Hence,
∫5tan(x)
cos2(x)dx =
∫5udu =
1
ln(5)5u + c
=1
ln(5)5tan(x) + c.
![Page 82: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/82.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Examples: Evaluate the following integrals.[1]∫ 0
−35xdx =
[1
ln(5)5x]0−3
=1
ln(5)
[1− 5−3
]=
124
125 ln(5).
[2] ∫5tan(x)
cos2(x)dx .
Put = tan(x), so du = sec2(x)dx = 1cos2(x)
dx . Hence,
∫5tan(x)
cos2(x)dx =
∫5udu =
1
ln(5)5u + c =
1
ln(5)5tan(x) + c.
![Page 83: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/83.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
[3]
∫(3x + 1)2
3xdx =
∫ ((3x)2 + 2(3x) + 1
)23x
dx
=
∫ [(3x)2
3x+
2(3x)
3x+
1
3x
]dx
=
∫ [3x + 2 + 3−x
]dx
=3x
ln(3)+ 2x − 3−x
ln(3)+ c .
![Page 84: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/84.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
[3] ∫(3x + 1)2
3xdx
=
∫ ((3x)2 + 2(3x) + 1
)23x
dx
=
∫ [(3x)2
3x+
2(3x)
3x+
1
3x
]dx
=
∫ [3x + 2 + 3−x
]dx
=3x
ln(3)+ 2x − 3−x
ln(3)+ c .
![Page 85: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/85.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
[3] ∫(3x + 1)2
3xdx =
∫ ((3x)2 + 2(3x) + 1
)23x
dx
=
∫ [(3x)2
3x+
2(3x)
3x+
1
3x
]dx
=
∫ [3x + 2 + 3−x
]dx
=3x
ln(3)+ 2x − 3−x
ln(3)+ c .
![Page 86: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/86.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
[3] ∫(3x + 1)2
3xdx =
∫ ((3x)2 + 2(3x) + 1
)23x
dx
=
∫ [(3x)2
3x+
2(3x)
3x+
1
3x
]dx
=
∫ [3x + 2 + 3−x
]dx
=3x
ln(3)+ 2x − 3−x
ln(3)+ c .
![Page 87: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/87.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
[3] ∫(3x + 1)2
3xdx =
∫ ((3x)2 + 2(3x) + 1
)23x
dx
=
∫ [(3x)2
3x+
2(3x)
3x+
1
3x
]dx
=
∫ [3x + 2 + 3−x
]dx
=3x
ln(3)+ 2x − 3−x
ln(3)+ c .
![Page 88: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/88.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
[3] ∫(3x + 1)2
3xdx =
∫ ((3x)2 + 2(3x) + 1
)23x
dx
=
∫ [(3x)2
3x+
2(3x)
3x+
1
3x
]dx
=
∫ [3x + 2 + 3−x
]dx
=3x
ln(3)
+ 2x − 3−x
ln(3)+ c .
![Page 89: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/89.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
[3] ∫(3x + 1)2
3xdx =
∫ ((3x)2 + 2(3x) + 1
)23x
dx
=
∫ [(3x)2
3x+
2(3x)
3x+
1
3x
]dx
=
∫ [3x + 2 + 3−x
]dx
=3x
ln(3)+ 2x
− 3−x
ln(3)+ c .
![Page 90: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/90.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
[3] ∫(3x + 1)2
3xdx =
∫ ((3x)2 + 2(3x) + 1
)23x
dx
=
∫ [(3x)2
3x+
2(3x)
3x+
1
3x
]dx
=
∫ [3x + 2 + 3−x
]dx
=3x
ln(3)+ 2x − 3−x
ln(3)
+ c .
![Page 91: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/91.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
[3] ∫(3x + 1)2
3xdx =
∫ ((3x)2 + 2(3x) + 1
)23x
dx
=
∫ [(3x)2
3x+
2(3x)
3x+
1
3x
]dx
=
∫ [3x + 2 + 3−x
]dx
=3x
ln(3)+ 2x − 3−x
ln(3)+ c .
![Page 92: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/92.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Definition
If a 6= 1 and y = ax ,
then the inverse function of y is denotedby loga(x) and is called logarithm function with base a. So
y = loga(x)⇐⇒x = ay .
Examples:
(i) 9 = 32 ⇐⇒ 2 = log3(9).
(ii) 25 = 52 ⇐⇒ 2 = log5(25).
(iii) 125 = 53 ⇐⇒ 3 = log5(125).
![Page 93: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/93.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Definition
If a 6= 1 and y = ax , then the inverse function of y
is denotedby loga(x) and is called logarithm function with base a. So
y = loga(x)⇐⇒x = ay .
Examples:
(i) 9 = 32 ⇐⇒ 2 = log3(9).
(ii) 25 = 52 ⇐⇒ 2 = log5(25).
(iii) 125 = 53 ⇐⇒ 3 = log5(125).
![Page 94: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/94.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Definition
If a 6= 1 and y = ax , then the inverse function of y is denotedby loga(x)
and is called logarithm function with base a. So
y = loga(x)⇐⇒x = ay .
Examples:
(i) 9 = 32 ⇐⇒ 2 = log3(9).
(ii) 25 = 52 ⇐⇒ 2 = log5(25).
(iii) 125 = 53 ⇐⇒ 3 = log5(125).
![Page 95: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/95.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Definition
If a 6= 1 and y = ax , then the inverse function of y is denotedby loga(x) and is called logarithm function with base a.
So
y = loga(x)⇐⇒x = ay .
Examples:
(i) 9 = 32 ⇐⇒ 2 = log3(9).
(ii) 25 = 52 ⇐⇒ 2 = log5(25).
(iii) 125 = 53 ⇐⇒ 3 = log5(125).
![Page 96: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/96.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Definition
If a 6= 1 and y = ax , then the inverse function of y is denotedby loga(x) and is called logarithm function with base a. So
y = loga(x)⇐⇒x = ay .
Examples:
(i) 9 = 32 ⇐⇒ 2 = log3(9).
(ii) 25 = 52 ⇐⇒ 2 = log5(25).
(iii) 125 = 53 ⇐⇒ 3 = log5(125).
![Page 97: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/97.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Definition
If a 6= 1 and y = ax , then the inverse function of y is denotedby loga(x) and is called logarithm function with base a. So
y = loga(x)⇐⇒x = ay .
Examples:
(i) 9 = 32
⇐⇒ 2 = log3(9).
(ii) 25 = 52 ⇐⇒ 2 = log5(25).
(iii) 125 = 53 ⇐⇒ 3 = log5(125).
![Page 98: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/98.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Definition
If a 6= 1 and y = ax , then the inverse function of y is denotedby loga(x) and is called logarithm function with base a. So
y = loga(x)⇐⇒x = ay .
Examples:
(i) 9 = 32 ⇐⇒ 2 = log3(9).
(ii) 25 = 52 ⇐⇒ 2 = log5(25).
(iii) 125 = 53 ⇐⇒ 3 = log5(125).
![Page 99: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/99.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Definition
If a 6= 1 and y = ax , then the inverse function of y is denotedby loga(x) and is called logarithm function with base a. So
y = loga(x)⇐⇒x = ay .
Examples:
(i) 9 = 32 ⇐⇒ 2 = log3(9).
(ii) 25 = 52
⇐⇒ 2 = log5(25).
(iii) 125 = 53 ⇐⇒ 3 = log5(125).
![Page 100: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/100.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Definition
If a 6= 1 and y = ax , then the inverse function of y is denotedby loga(x) and is called logarithm function with base a. So
y = loga(x)⇐⇒x = ay .
Examples:
(i) 9 = 32 ⇐⇒ 2 = log3(9).
(ii) 25 = 52 ⇐⇒ 2 = log5(25).
(iii) 125 = 53 ⇐⇒ 3 = log5(125).
![Page 101: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/101.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Definition
If a 6= 1 and y = ax , then the inverse function of y is denotedby loga(x) and is called logarithm function with base a. So
y = loga(x)⇐⇒x = ay .
Examples:
(i) 9 = 32 ⇐⇒ 2 = log3(9).
(ii) 25 = 52 ⇐⇒ 2 = log5(25).
(iii) 125 = 53
⇐⇒ 3 = log5(125).
![Page 102: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/102.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Definition
If a 6= 1 and y = ax , then the inverse function of y is denotedby loga(x) and is called logarithm function with base a. So
y = loga(x)⇐⇒x = ay .
Examples:
(i) 9 = 32 ⇐⇒ 2 = log3(9).
(ii) 25 = 52 ⇐⇒ 2 = log5(25).
(iii) 125 = 53 ⇐⇒ 3 = log5(125).
![Page 103: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/103.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Note:
(i)
loga(x) =ln(x)
ln(a).
Proof:
y = logxa ⇒ x = ay ⇒ ln(x) = ln(ay )
⇒ ln(x) = y ln(a)⇒ y =ln(x)
ln(a).
(ii)ln(x) = loge(x).
(iii)log(x) = log10(x)
.
![Page 104: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/104.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Note:
(i)
loga(x) =ln(x)
ln(a).
Proof:
y = logxa ⇒ x = ay ⇒ ln(x) = ln(ay )
⇒ ln(x) = y ln(a)⇒ y =ln(x)
ln(a).
(ii)ln(x) = loge(x).
(iii)log(x) = log10(x)
.
![Page 105: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/105.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Note:
(i)
loga(x) =ln(x)
ln(a).
Proof:
y = logxa
⇒ x = ay ⇒ ln(x) = ln(ay )
⇒ ln(x) = y ln(a)⇒ y =ln(x)
ln(a).
(ii)ln(x) = loge(x).
(iii)log(x) = log10(x)
.
![Page 106: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/106.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Note:
(i)
loga(x) =ln(x)
ln(a).
Proof:
y = logxa ⇒
x = ay ⇒ ln(x) = ln(ay )
⇒ ln(x) = y ln(a)⇒ y =ln(x)
ln(a).
(ii)ln(x) = loge(x).
(iii)log(x) = log10(x)
.
![Page 107: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/107.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Note:
(i)
loga(x) =ln(x)
ln(a).
Proof:
y = logxa ⇒ x = ay
⇒ ln(x) = ln(ay )
⇒ ln(x) = y ln(a)⇒ y =ln(x)
ln(a).
(ii)ln(x) = loge(x).
(iii)log(x) = log10(x)
.
![Page 108: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/108.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Note:
(i)
loga(x) =ln(x)
ln(a).
Proof:
y = logxa ⇒ x = ay ⇒
ln(x) = ln(ay )
⇒ ln(x) = y ln(a)⇒ y =ln(x)
ln(a).
(ii)ln(x) = loge(x).
(iii)log(x) = log10(x)
.
![Page 109: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/109.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Note:
(i)
loga(x) =ln(x)
ln(a).
Proof:
y = logxa ⇒ x = ay ⇒ ln(x) = ln(ay )
⇒ ln(x) = y ln(a)⇒ y =ln(x)
ln(a).
(ii)ln(x) = loge(x).
(iii)log(x) = log10(x)
.
![Page 110: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/110.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Note:
(i)
loga(x) =ln(x)
ln(a).
Proof:
y = logxa ⇒ x = ay ⇒ ln(x) = ln(ay )
⇒
ln(x) = y ln(a)⇒ y =ln(x)
ln(a).
(ii)ln(x) = loge(x).
(iii)log(x) = log10(x)
.
![Page 111: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/111.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Note:
(i)
loga(x) =ln(x)
ln(a).
Proof:
y = logxa ⇒ x = ay ⇒ ln(x) = ln(ay )
⇒ ln(x) = y ln(a)
⇒ y =ln(x)
ln(a).
(ii)ln(x) = loge(x).
(iii)log(x) = log10(x)
.
![Page 112: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/112.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Note:
(i)
loga(x) =ln(x)
ln(a).
Proof:
y = logxa ⇒ x = ay ⇒ ln(x) = ln(ay )
⇒ ln(x) = y ln(a)⇒
y =ln(x)
ln(a).
(ii)ln(x) = loge(x).
(iii)log(x) = log10(x)
.
![Page 113: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/113.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Note:
(i)
loga(x) =ln(x)
ln(a).
Proof:
y = logxa ⇒ x = ay ⇒ ln(x) = ln(ay )
⇒ ln(x) = y ln(a)⇒ y =ln(x)
ln(a).
(ii)ln(x) = loge(x).
(iii)log(x) = log10(x)
.
![Page 114: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/114.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Note:
(i)
loga(x) =ln(x)
ln(a).
Proof:
y = logxa ⇒ x = ay ⇒ ln(x) = ln(ay )
⇒ ln(x) = y ln(a)⇒ y =ln(x)
ln(a).
(ii)ln(x) = loge(x).
(iii)log(x) = log10(x)
.
![Page 115: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/115.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Note:
(i)
loga(x) =ln(x)
ln(a).
Proof:
y = logxa ⇒ x = ay ⇒ ln(x) = ln(ay )
⇒ ln(x) = y ln(a)⇒ y =ln(x)
ln(a).
(ii)ln(x) = loge(x).
(iii)log(x) = log10(x)
.
![Page 116: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/116.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 117: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/117.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 118: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/118.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).
Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 119: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/119.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 120: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/120.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′
=1
ln(a)
1
x.
2 y = log(ln(x)).
We have y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 121: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/121.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 122: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/122.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 123: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/123.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have
y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 124: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/124.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have y
= log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 125: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/125.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have y = log(ln(x))
= log10(ln(x)) = ln(ln(x))ln(10) . Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 126: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/126.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have y = log(ln(x)) = log10(ln(x))
= ln(ln(x))ln(10) . Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 127: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/127.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) .
Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 128: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/128.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 129: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/129.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,
y ′
=1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 130: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/130.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))]
=1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 131: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/131.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 132: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/132.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 133: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/133.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′
=1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 134: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/134.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)]
=1
log(x).
1
ln(10).1
x.
![Page 135: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/135.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find y ′ in each of the following:
1 y = loga(x).
We know that loga(x) = ln(x)ln(a) = 1
ln(a) ln(x).Thus,
y ′ =1
ln(a)
1
x.
2 y = log(ln(x)).
We have y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,
y ′ =1
ln(10)
d
dx[ln(ln(x))] =
1
ln(10)
1
ln(x)
1
x.
3 y = ln(log(x)).
y ′ =1
log(x)
d
dx[log(x)] =
1
log(x).
1
ln(10).1
x.
![Page 136: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/136.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find f ′(x) if f (x) = x4+x2 .
We take ln for both sides, we get
ln[f (x)] = ln[x4+x2
]= (4 + x2) ln(x).
Now we differentiate with respect to x ,
1
f (x)f ′(x) = (4 + x2)
d
dx[ln(x)] + ln(x)
d
dx[4 + x2]
= (4 + x2)1
x+ ln(x)(2x)
Therefore,
f ′(x) =
[4 + x2
x+ ln(x)(2x)
]x4+x2 .
![Page 137: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/137.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find f ′(x) if f (x) = x4+x2 .We take ln for both sides, we get
ln[f (x)] = ln[x4+x2
]= (4 + x2) ln(x).
Now we differentiate with respect to x ,
1
f (x)f ′(x) = (4 + x2)
d
dx[ln(x)] + ln(x)
d
dx[4 + x2]
= (4 + x2)1
x+ ln(x)(2x)
Therefore,
f ′(x) =
[4 + x2
x+ ln(x)(2x)
]x4+x2 .
![Page 138: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/138.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find f ′(x) if f (x) = x4+x2 .We take ln for both sides, we get
ln[f (x)]
= ln[x4+x2
]= (4 + x2) ln(x).
Now we differentiate with respect to x ,
1
f (x)f ′(x) = (4 + x2)
d
dx[ln(x)] + ln(x)
d
dx[4 + x2]
= (4 + x2)1
x+ ln(x)(2x)
Therefore,
f ′(x) =
[4 + x2
x+ ln(x)(2x)
]x4+x2 .
![Page 139: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/139.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find f ′(x) if f (x) = x4+x2 .We take ln for both sides, we get
ln[f (x)] = ln[x4+x2
]
= (4 + x2) ln(x).
Now we differentiate with respect to x ,
1
f (x)f ′(x) = (4 + x2)
d
dx[ln(x)] + ln(x)
d
dx[4 + x2]
= (4 + x2)1
x+ ln(x)(2x)
Therefore,
f ′(x) =
[4 + x2
x+ ln(x)(2x)
]x4+x2 .
![Page 140: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/140.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find f ′(x) if f (x) = x4+x2 .We take ln for both sides, we get
ln[f (x)] = ln[x4+x2
]= (4 + x2) ln(x).
Now we differentiate with respect to x ,
1
f (x)f ′(x) = (4 + x2)
d
dx[ln(x)] + ln(x)
d
dx[4 + x2]
= (4 + x2)1
x+ ln(x)(2x)
Therefore,
f ′(x) =
[4 + x2
x+ ln(x)(2x)
]x4+x2 .
![Page 141: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/141.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find f ′(x) if f (x) = x4+x2 .We take ln for both sides, we get
ln[f (x)] = ln[x4+x2
]= (4 + x2) ln(x).
Now we differentiate with respect to x ,
1
f (x)f ′(x)
= (4 + x2)d
dx[ln(x)] + ln(x)
d
dx[4 + x2]
= (4 + x2)1
x+ ln(x)(2x)
Therefore,
f ′(x) =
[4 + x2
x+ ln(x)(2x)
]x4+x2 .
![Page 142: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/142.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find f ′(x) if f (x) = x4+x2 .We take ln for both sides, we get
ln[f (x)] = ln[x4+x2
]= (4 + x2) ln(x).
Now we differentiate with respect to x ,
1
f (x)f ′(x) = (4 + x2)
d
dx[ln(x)] + ln(x)
d
dx[4 + x2]
= (4 + x2)1
x+ ln(x)(2x)
Therefore,
f ′(x) =
[4 + x2
x+ ln(x)(2x)
]x4+x2 .
![Page 143: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/143.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find f ′(x) if f (x) = x4+x2 .We take ln for both sides, we get
ln[f (x)] = ln[x4+x2
]= (4 + x2) ln(x).
Now we differentiate with respect to x ,
1
f (x)f ′(x) = (4 + x2)
d
dx[ln(x)] + ln(x)
d
dx[4 + x2]
= (4 + x2)1
x+ ln(x)(2x)
Therefore,
f ′(x) =
[4 + x2
x+ ln(x)(2x)
]x4+x2 .
![Page 144: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/144.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find f ′(x) if f (x) = x4+x2 .We take ln for both sides, we get
ln[f (x)] = ln[x4+x2
]= (4 + x2) ln(x).
Now we differentiate with respect to x ,
1
f (x)f ′(x) = (4 + x2)
d
dx[ln(x)] + ln(x)
d
dx[4 + x2]
= (4 + x2)1
x+ ln(x)(2x)
Therefore,
f ′(x)
=
[4 + x2
x+ ln(x)(2x)
]x4+x2 .
![Page 145: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/145.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Q: Find f ′(x) if f (x) = x4+x2 .We take ln for both sides, we get
ln[f (x)] = ln[x4+x2
]= (4 + x2) ln(x).
Now we differentiate with respect to x ,
1
f (x)f ′(x) = (4 + x2)
d
dx[ln(x)] + ln(x)
d
dx[4 + x2]
= (4 + x2)1
x+ ln(x)(2x)
Therefore,
f ′(x) =
[4 + x2
x+ ln(x)(2x)
]x4+x2 .
![Page 146: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/146.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Exercises: Find y ′ for the following:
1 y = ππ.
2 y = x4.
3 y = xπ.
4 y = πx .
5 y = x2x .
![Page 147: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/147.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Exercises: Find y ′ for the following:
1 y = ππ.
2 y = x4.
3 y = xπ.
4 y = πx .
5 y = x2x .
![Page 148: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/148.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Exercises: Find y ′ for the following:
1 y = ππ.
2 y = x4.
3 y = xπ.
4 y = πx .
5 y = x2x .
![Page 149: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/149.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Exercises: Find y ′ for the following:
1 y = ππ.
2 y = x4.
3 y = xπ.
4 y = πx .
5 y = x2x .
![Page 150: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/150.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Exercises: Find y ′ for the following:
1 y = ππ.
2 y = x4.
3 y = xπ.
4 y = πx .
5 y = x2x .
![Page 151: The Natural Logarithm and Exponential Functions · 2014. 1. 31. · Exponential Function Derivatives of General Exponential Function Integration of General Exponential Functions General](https://reader035.vdocument.in/reader035/viewer/2022062507/5fe5fa000038da5c213e9e9d/html5/thumbnails/151.jpg)
The NaturalLogarithm andExponentialFunctions
BanderAlmutairi
GeneralExponentialFunction
Derivatives ofGeneralExponentialFunction
Integration ofGeneralExponentialFunctions
GeneralLogarithmFunctions
Derivative ofGeneralLogarithmicFunction
Exercises: Find y ′ for the following:
1 y = ππ.
2 y = x4.
3 y = xπ.
4 y = πx .
5 y = x2x .