1. 3.6
The Chain Rule
Created by Greg Kelly, Hanford High School, Richland, Washington
Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts
Photo by Vickie Kelly, 2002

2. U.S.S. Alabama Mobile, Alabama
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2002

3. 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.

4. Consider a simple composite function:

5. and another:

6. This pattern is called the
and one more:
This pattern is called the
chain rule.
dy/dx = 2(3x + 1)1 • 3

7. Chain Rule: If is the composite of and , then:
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)

8. Chain Rule:
If is the composite of and , then:
Find:
example:

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

10. Here is a way to find the derivative by seeing “layers:”
Differentiate the outside function,
(keep the inner function unchanged...)
…then multiply by the derivative
of the inner function

11. Another example:
It looks like we need to use the chain rule again!
derivative of the
outside power
function
derivative of the
inside trig function

12. Another example:
The chain rule can be used more than once.
(That’s what makes the “chain” in the “chain rule”!)

13. Each derivative formula will now include the chain rule!
et cetera…

14. The derivative of x is one. derivative of outside function
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:
The derivative of x is one.
derivative of outside function
derivative of inside function

15. Don’t forget to use the chain rule!p