1. The Chain Rule By: Bryan Porter Caleb Clark Matt Devries

2. The Chain Rule Involves taking the derivative of a function with a different function inside of it To solve you need to: – Take the derivative of the outside – Leave the inside alone – Multiply it with the derivative of the inside – It sometimes has a cycle creating a “chain reaction”

3. Example Problems

4. Examples Find the Derivative of Sin(x ) 2

5. Examples Find the Derivative of Sin(x ) 2

6. Examples Find the Derivative of cos(x ) 3

7. Examples Find the Derivative of cos(x ) 3

8. Examples Find the Derivative of ln(x ) 2

9. Examples Find the Derivative of ln(x ) 2

10. Examples Find the Derivative of log (x ) 2 9

11. Examples Find the Derivative of log (x ) 2 9

12. Examples Find the Derivative of tan(x ) 4

13. Examples Find the Derivative of tan(x ) 4

14. Multiple Choice Questions

15. Multiple Choice Problem 1 What is the derivative of csc( X ) a. -cot(x )3x b. csc(x )cot(x )3x c. -csc(x )cot(x )3x d. cot(x )3x 3 3 33 33 3 2 2 2 2

16. Multiple Choice Problem 1 What is the derivative of csc( X ) a. -cot(x )3x b. csc(x )cot(x )3x c. -csc(x )cot(x )3x d. cot(x )3x 3 3 33 33 3 2 2 2 2

17. Multiple Choice Problem 2 What is the derivative of e a. e b. 4e c. e ln4 d. 4xe 4x

18. Multiple Choice Problem 2 What is the derivative of e a. e b. 4e c. e ln4 d. 4xe 4x

19. 9x x Multiple Choice Problem 3 What is the derivative of 3(ln(x )) a. b. c. d. 3 3x3x __ 3 3 3 3 2 2 9x9x 3x x

20. 9x x Multiple Choice Problem 3 What is the derivative of 3(ln(x )) a. b. c. d. 3 3x3x __ 3 3 3 3 2 2 9x9x 3x x

21. Multiple Choice Problem 4 Find the derivative of sin(cos(sin(x))) a. -cos(cos(sin(x)))sin(sin(x))cos(x) b. -cos(cos(sin(x)))sin(x)cos(x) c. cos(cos(sin(x))) d. -sin(sin(cos(x)))cos(cos(x))sin(x)

22. Multiple Choice Problem 4 Find the derivative of sin(cos(sin(x))) a. -cos(cos(sin(x)))sin(sin(x))cos(x) b. -cos(cos(sin(x)))sin(x)cos(x) c. cos(cos(sin(x))) d. -sin(sin(cos(x)))cos(cos(x))sin(x)

23. Multiple Choice Problem 5 What is the derivative of the ln(2 ) a. b. 2ln(2) c. d. none of the above 2x2x 1 x ___ 2 2x 2

24. Multiple Choice Problem 5 What is the derivative of the ln(2 ) a. b. 2ln(2) c. d. none of the above 2x2x 1 x ___ 2 2x 2

25. Free Response Question

26. Free Response Pocahontas is running through the woods in order to save John Smith from being killed by her father. At any time T ( in minutes) the distance x (hundred steps) between John and Pocahontas can be graphed by the function x=- Te +sin(T) +50 8 ( ) tan (T) _______

27. Free Response a. To the hundredth decimal place, how long does it take Pocahontas to reach John Smith?

28. b. If John Smith is being led away from Pocahontas at a steady rate of 100 steps per minute, say what Pocahontas’ average speed is as she races to save John Smith? Be sure to answer using correct units. Free Response

29. c. Find a formula v, in terms of T, that can be used to find Pocahontas’ instantaneous velocity during her race to save John Smith.

31. ( ) Free Response Solutions a. Set x=- Te +sin(T) +50 equal to 0. 8 When Solved T= 84.57 minutes tan (T) _______

32. Free Response Solutions b. the average speed is the starting distance, divided by the time that is spent.(slope of the secant line) and then add John Smith’s speed. The answer is about 269.14 steps per minute

33. Free Response Solutions c. You need to use the chain rule to find the derivative of the function x as seen below The answer becomes v= v=-1 Te +cos(T) ( ) tan (T) _______ T +1 2 8 __ +e tan (T)

34. For More Help… Visit http://archives.math.utk.edu/visual.calculus/2 /chain_rule.4/index.html http://archives.math.utk.edu/visual.calculus/2 /chain_rule.4/index.html Or if you do not have access to a computer, go talk to your calculus teacher