3.6 The Chain Rule

Photo by Vickie Kelly, 2002Created by Greg Kelly, Hanford High School, Richland, WashingtonRevised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002

U.S.S. AlabamaMobile, Alabama

We now have a pretty good list of “shortcuts” to find derivatives of simple functions.

Of course, many of the functions that we will encounter are not so simple. What is needed is a way to combine derivative rules to evaluate more complicated functions.

Consider a simple composite function:

6 10y x

3 52y x

nd ,5a 3u x 2( )y u

6 10y x 2( )y u 3 5u x

6dy

dx 2

dy

du 3

du

dx

dy

d

u

u

dy d

dx dx

6 32

( '( ) ( ) *')d

f g x g x xfx

gd

and another:

5 2y u

if 3u t

5 3 2( )y t

3u t

15dy

dt 5

dy

du 3

du

dt

15

5 3 2y t

15 2y t

5 2y u

dy

d

u

u

dy d

dx dx

( '( ) ( ) *')d

f g x g x xfx

gd

5 3

and one more:29 6 1y x x

23 1y x

3 1u x

3 1u x

18 6dy

xdx

2dy

udu

3du

dx

dy dy du

dx du dx

2y u

2when y u

29 6 1y x x

2 3 1dy

xdu

This pattern is called the chain rule.

dy/dx = 2(3x + 1)1 • 3

or ( ( )) ' ( ) * 'd

f g x f g x g xdx

dy dy du

dx du dx Chain Rule:

If is the composite of and , then:

f g y f u u g x

at at g x xf g f g

If f(g(x)) is the composite of y = f(u) and u = g(x), then:

d/dx(f(g(x)) = d/dx f (at g(x)) • d/dx g(at x)

dy dy du

dx du dx Chain Rule:

If is the composite of and , then:

f g y f u u g x

at at ( ) ( )( )xg x g xf g f g f g x

example:

sinf x x

Find:

cosf x x

2g x x 2 22g 0 2f g f g

1 4 4f g

' cos 10 (0)f

22 2 4 0g 2 4g x x

( )

2

g xf g f g

a x

x

t

( 22)gf g f g

We could also find the derivative at x = 2 this way:

2sin 4f g x x

2sin 4y x

siny u 2 4u x

cosdy

udu

2du

xdx

dy dy du

dx du dx

cos 2dy

u xdx

2cos 4 2dy

x xdx

2cos 2 4 2 2dy

dx

cos 0 4dy

dx

4dy

dx

Here is a way to find the derivative by seeing “layers:”

2sin 4y x

2 2' cos 4 4d

y x xdx

2' cos 4 2y x x

Differentiate the outside function,(keep the inner function unchanged...)

…then multiply by the derivative of the inner function

Evaluate this general derivative at 2, to find ' 4x y

Another example:

2cos 3d

xdx

2cos 3

dx

dx

2 cos 3 cos 3d

x xdx

derivative of theoutside power function

derivative of theinside trig function

It looks like we need to use the chain rule again!

Another example:

2cos 3d

xdx

2cos 3

dx

dx

2 cos 3 cos 3d

x xdx

2cos 3 sin 3 3d

x x xdx

2cos 3 sin 3 3x x

6cos 3 sin 3x x

The chain rule can be used more than once.

(That’s what makes the “chain” in the “chain rule”!)

Each derivative formula will now include the chain rule!

1n nd duu nu

dx dx sin cos

d duu u

dx dx

cos sind du

u udx dx

2tan secd du

u udx dx

et cetera…

The most common mistake in differentiating is to forget to use the chain rule.

Every derivative problem could be thought of as a chain-rule situation:

2dx

dx2

dx x

dx 2 1x 2x

derivative of outside function

derivative of inside function

The derivative of x is one.

Don’t forget to use the chain rule!