1. What we can do

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. The Chain Rule

4. Consider a simple composite function:

5. and another:

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

7. Some Baby Chain Rules:

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. Every derivative problem could be thought of as a chain-rule problem:
The most common mistake on the 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

11. What we don’t know yet Example 1 You will need a “u-substitution”
Let u = inner most function
Find u’
Rewrite y in terms of u
Find y’u
Multiply y’u*u’

12. Example 3 Derivative of outside function.
Derivative of inside function.

13. The Chain Rule Inside function is 2x+3 Outside Function is cos
Example:
Inside function is 2x+3
Outside Function is cos

14. The Chain Rule Rewrite as Outside function is Inside function is
Example:
Rewrite as
Outside function is
Inside function is

15. Peeling an Onion?? Example 4
Identify the “layers” of the composite function
Derive each layer from outside in (like peeling an onion)
Multiply all the derivatives together
Example 4

31. Try these:

32. Try these:

33. Try these:

34. Class Work:

35. Class Work - Answers: