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Sections 12.4 - 12.5Derivatives of Exponential and

Logarithmic Functions

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Derivative of Exponential Functions

( ) ( ) '( )f x f xd e e f xdx

General Form (Chain Rule)

x xd e edx

Basic Functions

lnx xd a a adx

( ) ( ) ln '( )f x f xd a a a f x

dx

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ExamplesFind dy/dx for the following functions.

28( )xy e x

7 5xy x 210 3 24 x xy e

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Examples

6 66 66

t ty tt

238 2 3 7t ty e t 35 1

3

t

tye

Find dy/dt for the function.

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Derivative of Logarithmic Functions

'( )ln ( )( )

d f xf xdx f x

General Form (Chain Rule)

1lnd xdx x

Basic Functions

1loglna

d xdx x a

'( )log ( )( ) lna

d f xf xdx f x a

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ExamplesFind dy/dx for the following functions.

32(ln 5)y x

38log (2 5)y x x

3 2log xy x e

35log (3 1 2 )xy x e

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Examples

7(5 3)lnxy

x

(3 5 ) 2 3x xy e x

(4 ) ln(3 1)xy x e x

Find y’ for the function.

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Example (#38 on page 781)

( ) 900 800 1.1 xC x

The cost in dollars to produce x DVDs can be approximated by.

Find and interpret the marginal cost when the following quantities are madea) 0b) 20c) What happens to the marginal cost as the

number produced becomes larger and larger?d) Find the average cost for producing 30 DVDs