Chapter 4Calculating the

Derivative

JMerrill, 2009

Review

Find the derivative of (3x – 2x2)(5 + 4x)

-24x2 + 4x + 15 Find the derivative of

25x 2x 1

22 2

5x 4x 5(x 1)

4.3The Chain Rule

Composition of Functions

A composition of functions is simply putting 2 functions together—one inside the other.

Example: In order to convert Fahrenheit to Kelvin we have to use a 2-step process by first converting Fahrenheit to Celsius.

89oF = 31.7oC 31.7oC = 304.7K But if we put 1 function inside the other function,

then it is a 1-step process.

5C (F 32) 9K C 273

Composition of Functions

We are used to writing f(x). f(g(x)) simply means that g(x) is our new x in the f equation.

We can also go the other way. means g(f(x)).

The composite of f(x) and g(x) is denoted which means the same as f(g(x)).

f g x

g f x

Given 2( ) 4 2 ( ) 2f x x x g x x

f(g(3)) =

= f(6)

= 4(6)2 – 2(6)

= 144 – 12

= 132

g(3) = 6

Given 1( ) ( ) 1f x g x xx

f g x

( ( ))f g x ( ( ))( 1)f g xf x

( ( ))( 1)1

1

f g xf x

x

g f x

( ( ))1

1 1

g f x

gx

x

g(x) = x+1

Substitute x+1In place of the

x in the f equation

=

The new x in the g equation

The Chain Rule

Chain Rule Example

Use the chain rule to find Dx(x2 + 5x)8

Let u = x2 + 5x Let y = u8

72

7

dy dy dudx du dx 8u 2

8 x 5x

x 5

2x 5

Another way to think of it: The derivative of the outside times the derivative of the inside

Chain Rule – You Try

Use the chain rule to find Dx(3x - 2x2)3

Let u = 3x - 2x2 Let y = u3

22

2

dy dy dudx du dx 3u 3 4x

3x 2x 3 4x3

The derivative of the outside times the derivative of the inside

Chain Rule

Find the derivative of y = 4x(3x + 5)5

This is the Product Rule inside the Chain Rule. Let u = 3x + 5; y = u5

4

4 5

4 5

4 5

5

4x 5u (3) (3x 5) (4)

4x 5(3x 5) (3) 4(3x 5)

4x 15(3x 5) 4(360x(3x 5)

x 5)4(3x 5)

Chain Rule

4

4

4

5

Factor out the common f actor14(3x 5)

60x(3x 5) 4(3x 5)

4(3x 5) (18x 55x (3x 5)

)

Chain Rule

Find the derivative of This is the Quotient Rule in the Chain Rule Let u = 3x + 2; let y = u7

73x 2x 1

6 7

2

6

2

7

2

6 7

(x 1) 7u (3) (3x 2) (1)(x 1)

(x 1) 7(

21 (x 1)(3x 2) (3x 2

3x 2) (3) (3x 2)(x 1)

)(x 1)

Chain Rule

6

6 7

2

6

2

2

2

6

Factor out the common f actor21(x 1) (3x 2)(x

(3x 2)

(

21 (x 1)(3x 2) (3x 2)(x 1)

(3x 2)

3

1

x

8

1)21x 21 3x 2(x 1)

x 2)

2

3

)

(x 1

4.4 Derivatives of Exponential Functions

Derivative of ex

Derivative of ax

xx

x(lD 3 n3)3

Other Derivatives

Examples – Find the Derivative

y = e5x

g(x)

x 5x5e (g'(x)e (5) 5e

Examples – Find the Derivative

y = 32x+1

g(x)

2

2x 1

x 1

lna a g'(x)

ln3 32ln3 3

(2)

Example

Find if Use the product rule

1

21 5x 2 (5)2

52 5x 2

2x 1y e 5x 2 dydx

12 2x 1 x 12x xy e D 5x 2 5x 2 D e

2x 1e (2x)

Example

12 2x 1 x 12x x

2 2x 1 x 1

2x 1 2x 1

y e D 5x 2 5x 2 D e

5e 5x 2 2xe2 5x 25e 5x 2 2xe2 5x 2

Example Continued

2x 1 2x 1

2x 1 2

2 2x 1 x 1

2x 1

5e 2xe 5x 22 5x 2

5e e (4x)(5x 2)2 5x

e 2

2e 5 4x(5x 2)

2 5x

2 5x 2

0x 8x 52 5x

2 5 2

2

x

2

Get a common denominator to add the 2 parts together

4.5Derivatives of Logarithmic Functions

Definition

Bases – a side note Everything we do is in Base 10.

We count up to 9, then start over. We change our numbering every 10 units. 1 11 212 12 223 13 23…4 145 156 167 178 189 1910 20

Ones Place

One group of ten and 1, 2, 3…

ones

Two tens

and …ones

Bases The Yuki of Northern California used Base 8.

They counted up to 7, then started over. The numbering changed every 8 units.

1 13 252 14 263 15 27…4 165 176 207 2110 2211 2312 24

Ones Place

One eight

and 3…ones

Two eights and …ones

So, 17 in Base 8 = 15 in Base 10

258 = 2 eights + 5 ones = 21

Bases

The Mayans used Base 20. The Sumerians and people of Mesopotamia

used Base 60.

Definition

Example

Find f’(x) if f(x) = ln 6x Remember the properties of logs ln 6x = ln 6 + ln x

d d(ln6) (lnx)dx dx10 1

xx

Definitions

Examples – Find the Derivatives

y = ln 5x If g(x) = 5x, then g’(x) = 5

dy g'(x) 5dx g(x) 5

1xx

F’(x)

f(x) = 3x ln x2

Product Rule

2 2

2

2

2

df ' (x) (3x) lnx lnx (3)

6 3lnx

dx2x3x lnx (3)x