2.4 The Chain Rule

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

2 3 5y x

If 3 5u x

then 2y u

6 10y x 2y u 3 5u x

6dy

dx 2

dy

du 3

du

dx

dy dy du

dx du dx

6 2 3

and another:

5 2y u

where 3u t

then 5 3 2y t

3u t

15dy

dt 5

dy

du 3

du

dt

dy dy du

dt du dt

15 5 3

5 3 2y t

15 2y t

5 2y u

and one more:29 6 1y x x

23 1y x

If 3 1u x

3 1u x

18 6dy

xdx

2dy

udu

3du

dx

dy dy du

dx du dx

2y u

2then y u

29 6 1y x x

2 3 1dy

xdu

6 2dy

xdu

18 6 6 2 3x x This pattern is called the chain rule.

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 xu g xf g f g

example: sinf x x 2 4g x x Find: at 2f g x

cosf x x 2g x x 2 4 4 0g

0 2f g

cos 0 2 2

1 4 4

We could also do it 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 faster way to find the derivative:

2sin 4y x

2 2cos 4 4d

y x xdx

2cos 4 2y x x

Differentiate the outside function...

…then the inside function

At 2, 4x y

Here’s another

Now plug in u and simplify

Another example:

2cos 3d

xdx

2cos 3

dx

dx

2 cos 3 cos 3d

x xdx

derivative of theoutside function

derivative of theinside 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”!)

Derivative formulas 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

etcetera…

The most common mistake on the chapter 2 test is to forget to use the chain rule.

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

2dx

dx2d

x xdx

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!

HW Pg. 138 7-29 odd, 39-53, 91, 93, 102