1. The chain rule (2.4) October 23rd, 2012

2. I. the chain rule Thm. 2.10: 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 -or- Thm. 2.10: 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 -or-

3. Ex. 1: For each function, find the derivative by using the rules learned previously in sections 2.2 and 2.3. Then find the derivative using the chain rule. Which method is easier? a. b. c. Ex. 1: For each function, find the derivative by using the rules learned previously in sections 2.2 and 2.3. Then find the derivative using the chain rule. Which method is easier? a. b. c.

4. II. The general power rule Thm. 2.11: The General Power Rule: If, where u is a differentiable function of x and n is a rational number, then -or- Thm. 2.11: The General Power Rule: If, where u is a differentiable function of x and n is a rational number, then -or-

5. A. Applying the general power rule Ex. 2: Find the derivative of.

6. You Try: Find the derivative of.

7. B. Differentiating functions involving radicals Ex. 3: Find all the points on the graph of for which f ’(x) = 0 and those for which f ‘(x) does not exist.

8. You Try: Find all the points on the graph of for which f ‘(x) = 0 and those for which f ‘(x) does not exist.

9. C. differentiating quotients with constant numerators Ex. 4: Differentiate.

10. You Try: Differentiate.

11. III. Simplifying derivatives Ex. 5: Find the derivative of.

12. You Try: Find the derivative of.

13. Ex. 6: Find the derivative of.

14. You Try: Find the derivative of.

15. Ex. 7: Find the derivative of.

16. You Try: Find the derivative of.

17. IV. trigonometric functions & the chain rule A. Applying the Chain Rule to Trigonometric Functions Ex. 8: Find the derivative of each function. a. y = cos 2x b. y = sin (4x+1) c. y = cot 4x A. Applying the Chain Rule to Trigonometric Functions Ex. 8: Find the derivative of each function. a. y = cos 2x b. y = sin (4x+1) c. y = cot 4x

18. You Try: Find the derivative of each function. a. b. y = csc 4x + 1 c. d. You Try: Find the derivative of each function. a. b. y = csc 4x + 1 c. d.

19. B. repeated application of the chain rule Ex. 9: Find the derivative of.

20. You Try: Find the derivative of.

21. C. tangent line of a trigonometric function Ex. 10: Find an equation of the tangent line to the graph of at the point. Ex. 10: Find an equation of the tangent line to the graph of at the point.