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

Section 3.6 Derivatives of Log Functions2

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

Section 3.6 Derivatives of Log Functions3

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

Section 3.6 Derivatives of Log Functions4

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

Section 3.6 Derivatives of Log Functions5

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

Section 3.6 Derivatives of Log Functions6

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

Section 3.6 Derivatives of Log Functions7

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

Section 3.6 Derivatives of Log Functions8

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

Section 3.6 Derivatives of Log Functions9

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

Section 3.6 Derivatives of Log Functions10

Find the first and second derivatives.

Try these…

2

2

1. ln(2 )

2. ln 1

y x x

y x x

Section 3.6 Derivatives of Log Functions11

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

Try these…

2

ln (1,1)xy xe