1. Section 2.4 – The Chain Rule

2. Warm-Up Explain why we can not differentiate the function below:
Wrong answer: The exponent of 30.
Although the exponent makes the derivative difficult, we could use the product rule to find the derivative.
Right answer: The function inside of the secant function.
We have not taken the derivative of a composition of functions.

3. Composition of Functions
If and , find
COMPOSITION OF FUNCTIONS

4. Decomposition of Functions
If each function below represents , define and
DECOMPOSITION OF FUNCTIONS

5. The Chain Rule
If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f(g(x)) is a differentiable function of x, and
Other ways to write the Rule:

6. Instructions for The Chain Rule
For , to find :
Decompose the function
Differentiate the MOTHER FUNCTION
Differentiate the COMPOSED FUNCTION
Multiply the resultant derivatives
Substitute for u and Simplify
Make sure each function can be differentiated.

7. Example 1 Find if and . Define f and u:
Find the derivative of f and u:
Use the Chain Rule:
Substitute:
Substitute for u:
Simplify:

8. Example 2 Differentiate . Define f and u:
Find the derivative of f and u:
Use the Chain Rule:
Substitute:
Substitute for u:
Simplify:

9. Example 3 If f and g are differentiable, , , and ; find .
Define h and u:
Find the derivative of h and u:

10. Example 4
Find if
Define f and u:
Find the derivative of f and u:

11. White Board Challege
Find f '(-2) if:

12. Example 5 Differentiate . Define f and u:
Find the derivative of f and u: OR

13. Example 6 Differentiate . Use the old derivative rules
Chain Rule Twice

14. Example 7 Find the derivative of the function . Quotient Rule
Chain Rule

15. Example 8
Differentiate
Product Rule
Chain Rule Twice

16. White Board Challege
Find the equation of the tangent to the curve y=3sin(2x) at the point:

17. Example 9
Differentiate
Chain Rule
Chain Rule Again

18. Example 10 Find an equation of the tangent line to at .
Find the Derivative
Evaluate the Derivative at x = π
Find the equation of the line