1. Section 2.4 – The Chain Rule

2. Example 1 If and, find. COMPOSITION OF FUNCTIONS

3. Example 2 If each function below represents, define and. DECOMPOSITION OF FUNCTIONS

4. 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:

5. 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.

6. Example 1 Find if and. Define f and u: Find the derivative of f and u:

7. Example 2 Differentiate. Define f and u: Find the derivative of f and u:

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

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

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

11. Now try the Chain Rule in combination with all of our other rules.

12. Example 1 Differentiate. Chain Rule Twice Use the old derivative rules

13. Example 2 Find the derivative of the function. Chain Rule Quotient Rule

14. Example 3 Differentiate. Chain Rule Twice

15. Example 4 Differentiate. Chain Rule Chain Rule Again

16. Example 5 Find an equation of the tangent line to at. Find the Derivative Evaluate the Derivative at x = π Find the equation of the line