3.6 derivatives of logarithmic functions 1section 3.6 derivatives of log functions

Section 3.6 Derivatives of Log Functions1

Chapter 3 – Differentiation Rules

3.6 Derivatives of Logarithmic Functions


In this section we use implicit differentiation to find the derivatives of the logarithmic functions y = logax and, in particular, the natural logarithmic function y = ln x. [It can be proved that logarithmic functions are differentiable; this is certainly plausible from their graphs.

Derivatives of Log Functions


Properties of logs.

If a and b are positive numbers and n is rational, then the following properties are true:

Remember…

1.ln(1) 0

2.ln( ) ln lnab a b

3.ln( ) ln

4.ln ln ln

na n a

aa b

b


With Chain Rule

Definitions

1log

ln

1ln

1ln

a

dx

dx x a

dx

dx x

dx

dx x

1ln

d duu

dx u dx


Differentiate the functions.

Examples

2

5

10

1. log 1 3

2. ln

1 ln3.

1 ln

4. 2 log

f x x

g x x

tf t

t

k x x x


The calculation of complicated functions involving products, quotients, or powers can often be simplified by taking logarithms. This method is called logarithmic differentiation.

Logarithmic Differentiation


Take the natural log of both sides of the equation.

Use the Laws of Logs to simplify. Differentiate implicitly with respect to x. Solve for dy/dx.

Steps in Log Differentiation


Use log differentiation to find the derivative of the functions below.

Examples

2 102

cos

1. 1

2. ln

x

x

y xe x

y x


Differentiate the functions below.

Try these

2

1. ( ) sin ln(5 )

2. ( ) ln 1

3. ( ) ln 1

ln4. ( )

1 ln(2 )

y

f x x x

g x x x

F y y e

uf u

u

2

2

2

5. ln 2 5

6. ln 1

7. log cos

8.

x

x

y x

y x x

y e

y e x

x y


Find the first and second derivatives.

Try these…

2

2

1. ln(2 )

2. ln 1

y x x

y x x


Find an equation of the normal and tangent lines to the curve at the given point.

Try these…

2

ln (1,1)xy xe

3.6 derivatives of logarithmic functions 1section 3.6 derivatives of log functions

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