the chain rule
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/1.jpg)
The Chain RuleBy: Bryan Porter
Caleb ClarkMatt Devries
![Page 2: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/2.jpg)
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”
![Page 3: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/3.jpg)
Example Problems
![Page 4: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/4.jpg)
Examples
• Find the Derivative of Sin(x )2
![Page 5: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/5.jpg)
Examples
• Find the Derivative of Sin(x )2
![Page 6: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/6.jpg)
Examples
• Find the Derivative of cos(x )3
![Page 7: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/7.jpg)
Examples
• Find the Derivative of cos(x )3
![Page 8: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/8.jpg)
Examples
• Find the Derivative of ln(x )2
![Page 9: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/9.jpg)
Examples
• Find the Derivative of ln(x )2
![Page 10: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/10.jpg)
Examples
• Find the Derivative of log (x )29
![Page 11: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/11.jpg)
Examples
• Find the Derivative of log (x )29
![Page 12: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/12.jpg)
Examples
• Find the Derivative of tan(x )4
![Page 13: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/13.jpg)
Examples
• Find the Derivative of tan(x )4
![Page 14: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/14.jpg)
Multiple Choice Questions
![Page 15: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/15.jpg)
Multiple Choice Problem 1
• What is the derivative of csc(X )a. -cot(x )3xb. csc(x )cot(x )3xc. -csc(x )cot(x )3xd. cot(x )3x
3
3
3 3
3 3
3
2
2
2
2
![Page 16: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/16.jpg)
Multiple Choice Problem 1
• What is the derivative of csc(X )a. -cot(x )3xb. csc(x )cot(x )3xc. -csc(x )cot(x )3xd. cot(x )3x
3
3
3 3
3 3
3
2
2
2
2
![Page 17: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/17.jpg)
Multiple Choice Problem 2
• What is the derivative of ea. eb. 4ec. e ln4d. 4xe
4x
4x
4x
4x
4x
![Page 18: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/18.jpg)
Multiple Choice Problem 2
• What is the derivative of ea. eb. 4ec. e ln4d. 4xe
4x
4x
4x
4x
4x
![Page 19: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/19.jpg)
9xx
Multiple Choice Problem 3
• What is the derivative of 3(ln(x ))a.
b.
c.
d.
3
3x
__3
__
__
__
3
3
3
2
2
9x
3xx
![Page 20: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/20.jpg)
9xx
Multiple Choice Problem 3
• What is the derivative of 3(ln(x ))a.
b.
c.
d.
3
3x
__3
__
__
__
3
3
3
2
2
9x
3xx
![Page 21: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/21.jpg)
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)
![Page 22: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/22.jpg)
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)
![Page 23: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/23.jpg)
Multiple Choice Problem 5
• What is the derivative of the ln(2 )a.
b. 2ln(2)c.
d. none of the above
2x
1x
___
___
2
2x
2
![Page 24: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/24.jpg)
Multiple Choice Problem 5
• What is the derivative of the ln(2 )a.
b. 2ln(2)c.
d. none of the above
2x
1x
___
___
2
2x
2
![Page 25: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/25.jpg)
Free Response Question
![Page 26: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/26.jpg)
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
![Page 27: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/27.jpg)
Free Response
a. To the hundredth decimal place, how long does it take Pocahontas to reach John Smith?
![Page 28: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/28.jpg)
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
![Page 29: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/29.jpg)
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.
![Page 30: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/30.jpg)
![Page 31: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/31.jpg)
( )Free Response Solutions
a. Set x=- Te +sin(T) +50 equal to 0. 8
When Solved T= 84.57 minutes
tan (T)_______
-1
![Page 32: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/32.jpg)
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 about269.14 steps per minute
![Page 33: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/33.jpg)
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
_______T +128
__ +etan (T)
-1
![Page 34: The Chain Rule](https://reader036.vdocuments.us/reader036/viewer/2022062518/5681458c550346895db27549/html5/thumbnails/34.jpg)
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