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The Chain Rule

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

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The Chain Rule

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  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 2 3 3 3 2 3 3 2 2 3

  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 2 3 3 3 2 3 3 2 2 3

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

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

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

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

  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 2 • What is the derivative of the ln(2 ) a. b. 2ln(2) c. d. none of the above ___ 2 x 2 ___ 1 x 2x

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

  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) -1 _______

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

  28. Free Response 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.

  29. Free Response 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.

  30. Free Response Solutions ( ) -1 a. Set x=- Te +sin(T) +50 equal to 0. 8 When Solved T= 84.57 minutes tan (T) _______

  31. 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

  32. 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) -1 -1 tan (T) __ _______ +e 8 T +1 2

  33. For More Help… • Visit 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

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