3.6 The Chain Rule

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

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

2 3 5y x

If 3 5u x

then 2y u

6 10y x 2y u 3 5u x

6dydx

2dydu

3dudx

dy dy dudx du dx

6 2 3

and another:

5 2y u

where 3u t

then 5 3 2y t

3u t

15dydt

5dydu

3dudt

dy dy dudt 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

3dudx

dy dy dudx 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 dudx 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 dudx du dx

cos 2dy u xdx

2cos 4 2dy x xdx

2cos 2 4 2 2dydx

cos 0 4dydx

4dydx

Here is a faster way to find the derivative:

2sin 4y x

2 2cos 4 4dy x xdx

2cos 4 2y x x

Differentiate the outside function...

…then the inside function

At 2, 4x y

Another example:

2cos 3d xdx

2cos 3d x

dx

2 cos 3 cos 3dx 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 3d x

dx

2 cos 3 cos 3dx xdx

2cos 3 sin 3 3dx 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 nudx dx

sin cosd duu udx dx

cos sind duu udx dx

2tan secd duu udx dx

etcetera…

The formulas on the memorization sheet are written with instead of . Don’t forget to include the term!

uudu

dx

The most common mistake on the chapter 3 test is to forget to use the chain rule.

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

2d xdx

2 dx xdx

2 1x 2x

derivative of outside function

derivative of inside function

The derivative of x is one.

The chain rule enables us to find the slope of parametrically defined curves:

dy dy dxdt dx dt

dydydt

dx dxdt

Divide both sides bydxdtThe slope of a parametrized

curve is given by:

dydy dt

dxdxdt

These are the equations for an ellipse.

Example: 3cosx t 2siny t

3sindx tdt

2cosdy tdt

2cos3sin

dy tdx t

2 cot3

t

Don’t forget to use the chain rule!