11.4 The Chain Rule

Composite Functions

Definition: A function m is a composite of functions f and g if

m(x) = f [g(x)]

The domain of m is the set of all numbers x such that x is in the domain of g and g(x) is in the domain of f.

Example 1

Given f(u) = 2u and g(x) = ex

Find f [g(x)] and g [f(u)]

f [g(x)] = f (ex) substitute g(x) for ex

= 2(ex) plug ex into function f(u)

= 2ex

g [f(u)] = g (2u) substitute f(u) for 2u

= e2u plug 2u into function g(x)

Example 2

Write each function as a composite of two simplerfunctions

A) Y = 50 e-2x

g(x) = -2x and f(u) = 50 eu

B)

and

3 31 xy

31)( xxg 3)( uuf

C) Write y as a composite of two simpler functions

y = (2x + 1)100

g(x) = 2x + 1 and f(u) = u 100

How do we find the derivative of a composite

function such as the one above?

What is the derivative of (2x + 1)100

Let’s investigate on the next slide!

Find the derivative of (2x + 1)2, (2x + 1)3, (2x + 1)4 and (2x + 1)100

Use the product rule:F(x) = (2x + 1)2 = (2x + 1) (2x + 1)F’(x) = (2)(2x+1) + (2)(2x+1) = 4(2x+1)

F(x) = (2x + 1)3 = (2x + 1) (2x + 1)2 = (2x+1)(4x2 + 4x +1)F’(x) = (2)(4x2 + 4x +1) + (8x + 4)(2x + 1) = 2 (2x+1)2 + 4(2x+1)(2x+1) = 2 (2x+1)2 + 4(2x+1)(2x+1) = 2 (2x+1)2 + 4(2x+1)2 = 6 (2x+1)2 Try (2x + 1)4 you should find out that f’(x) = 8 (2x+1)3

Look at the relationship between f(x) and f’(x), do you agree that the derivative of (2x + 1)100 is 200 (2x+1)99 ?

f’(x)f(x)

4(2x+1) = 2(2x+1) · 2(2x + 1)2

n·[g(x)]n-1 · g’(x)Any composite function [g(x)] n

200(2x+1)99 = 100(2x+1)99 · 2(2x + 1)100

8(2x+1)3 = 4(2x+1)3 · 2(2x + 1)4

6(2x+1)2 = 3(2x+1)2 · 2(2x + 1)3

Generalized Power Rule

What we have seen is an example of the generalized power rule: If u is a function of x, then

1n nd duu nu

dx dx

Chain Rule

Chain Rule: If y = f (u) and u = g(x) define the composite function y = m(x) = f [g(x)], then

Or m’(x) = f’[g(x)] g’(x) provided that f’[g(x) and g’(x) exist

.existandprovided,dx

du

du

dy

dx

du

du

dy

dx

dy

We have used the generalized power rule to find derivatives of composite functions of the form f (g(x)) where f (u) = un is a power function. But what if f is not a power function? It is a more general rule, the chain rule, that enables us to compute the derivatives of many composite functions of the form f(g(x)).

Example 3Find the indicated derivatives

A) h’(x) if h(x) = (5x + 2)3

h’(x) = 3 (5x + 2)2 (5x+2)’

= 3 (5x + 2)2 (5) =15 (5x + 2)2

B) y’ if y = (x4 – 5)5

y’= 5(x4 – 5)4 (x4 – 5)’

= 5(x4 – 5)4 (4x3) = 20x3 (x4 – 5)4

Example 3Find the indicated derivatives

C)

D)

22 4

1

tdt

d 22 4

tdt

d )2(4232 tt

32 4

4

t

t

ifdw

dgwwg 4)( 2

1

)4( w

142

1 2

1

wdw

dg

2

1

42

1

w

Example 4

Find dy/du, du/dx, and dy/dx (express dy/dx as a function of x)

A) y = u-5 and u = 2x3 + 4

6

230

u

x

66 5

5u

udu

dy

26xdx

du

dx

du

du

dy

dx

dy )6(

5 26

xu

63

2

)42(

30

x

x

The problem above can be solved by taking the derivative of y = (2x3 + 4) -5

Example 4 (continue)

Find dy/du, du/dx, and dy/dx (express dy/dx as a function of x)

B) y = eu and u = 3x4 + 6

uex312

uedu

dy

312xdx

du

dx

du

du

dy

dx

dy )12( 3xeu 633 4

12 xex

This problem can be solved by taking the derivative of63 4 xey

Example 5

Find dy/du, du/dx, and dy/dx (express dy/dx as a function of x)

C) y = ln u and u = x2 + 9x + 4

udu

dy 1

92 xdx

du

dx

du

du

dy

dx

dy )92(

1 x

u 49

922

xx

x

This problem can be solved by taking the derivative of y = ln (x2 + 9x + 4)

The chain rule can be extended to compositions

of three or more functions.

Given: Y = f(w), w = g(u), and u = h(x)

Then

dx

du

du

dw

dw

dy

dx

dy

Example 6

y = h(x) = [ ln(1 + ex) ]3 find dy/dx

Let y = w 3, w = ln u, and u = 1 + ex

then

xeu

u1

)(ln3 2

dx

du

du

dw

dw

dy

dx

dy

xeu

wdx

dy

13 2

xx

x ee

e

1

1)]1[ln(3 2

x

xx

e

ee

1

)]1[ln(3 2

Example 6 (continue)

y = h(x) = [ ln(1 + ex) ]3 find dy/dx

Another way

)(1

1)1ln(3

2 xx

x ee

edx

dy

x

xx

e

ee

1

)]1[ln(3 2

Derivative of the outside function

The derivative of the function inside the [ ]

Derivative of the function inside ( )

Examples for the Power Rule

Chain rule terms are marked:

?',)(ln

?',)(

?',)12(

?',)2(

?',)2(

?',

5

5

5

53

5

5

yxy

yey

yxy

yxy

yxy

yxy

x

Compare with the next slide

Examples for the Power Rule

Chain rule terms are marked:

x

xxxyxy

eeeyey

xxyxy

xxxyxy

xxyxy

xyxy

xxxx

445

545

445

1424353

445

45

)(ln5)/1()(ln5',)(ln

5)()(5',)(

)12(10)2()12(5',)12(

480)6()2(5',)2(

160)2()2(5',)2(

5',

Examples for Exponential Derivatives

dx

duee

dx

d uu

?',

?',

?',

?',

ln

534

13

3

2

yey

yey

yey

yey

x

xx

x

x

Compare with the next slide

Examples for Exponential Derivatives

dx

duee

dx

d uu

1)1

(',

)38(',

3)3(',

3)3(',

lnlnln

534534

131313

333

22

x

x

x

e

xeyey

xeyey

eeyey

eeyey

xxx

xxxx

xxx

xxx

Property of ln

Examples for Logarithmic Derivatives

dx

du

uu

dx

d

1ln

?'),42ln(

?'),ln(

?'),14ln(

?'),4ln(

2

2

yxxy

yxy

yxy

yxy

Compare with the next slide

Examples for Logarithmic Derivatives

dx

du

uu

dx

d

1ln

42

22)22(

42

1'),42ln(

2)2(

1'),ln(

14

44

14

1'),14ln(

14

4

1'),4ln(

222

22

xx

xx

xxyxxy

xx

xyxy

xxyxy

xxyxy