1. AP Calculus AB
4.1 The Chain Rule

2. 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.

3. Consider a simple composite function:

4. and another:

5. and one more:
This pattern is called the chain rule.

6. Example:
Find:

7. Here is a faster way to find the derivative:
Differentiate the outside function...
…then the inside function
This is called the “Outside-Inside” Rule. See pg. 150

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

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

10. Derivative formulas include the chain rule!
etcetera…

11. Every derivative problem could be thought of as a chain-rule problem:
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:
The derivative of x is one.
derivative of outside function
derivative of inside function

12. The chain rule enables us to find the slope of parametrically defined curves:
Divide both sides by
The slope of a parametrized curve is given by:

13. Example:
These are the equations for an ellipse.

14. # 13, pg. 153
derivative of the
outside function
derivative of the
inside function

15. # 15, pg. 153
Sometimes it is helpful to rewrite the function so you can see the outside and inside functions better.

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