Section 3.4The Chain Rule

• Consider the function– We can “decompose” this function into two

functions we know how to take the derivative of– For example

– What are

– Now think of as

32 )3( xy

3)(and)( 23 xxguuufy

dx

du

du

dyand

dx

du

du

dy

dx

dy

dx

dy

Theorem 3.5: The Chain Rule

• Let g and f be differentiable functions then

• The derivative of a composite function is the derivative of the outside function (leaving the inside function alone) times the derivative of the inside function

)('))(('))'((

)('))(('))((

xgxgfxgf

xgxgfxgfdx

d

Consider ___________

OUTSIDE FUNCTION:

INSIDE FUNCTION:

So we have:

10322103210 )7(333)7(11311 xxxxxudx

du

du

dy

10

11

11udu

dy

uy

23 37 xdx

duxu

113 )7( xy

73 xu

The General Power Rule

1)]([)( nn xgnxgdx

d

The General Exponential Rule

( ) ( ) '( )f x f xde e f x

dx

Examples

3

2

3

3 2 5

3

1002 3

2

3

( )

( ) ( )

( )

3

( )7 3

( ) 1

x

x

x

x

f x e

s x x x

h x xe

y x e

xq x

x

g x e