3.6

1. Photo by Vickie Kelly, 2002 Greg Kelly, Hanford High School, Richland, Washington The Chain Rule 3.6

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

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 chain rule. and one more:

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

8. We could also do it this way:

9. Here is a faster way to find the derivative: Differentiate the outside function... …then the inside function

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

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

12. The formulas on the memorization sheet are written with instead of . Don’t forget to include the term! Derivative formulas include the chain rule! etcetera…

13. 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

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

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

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