The Chain RuleBy: Bryan Porter
Caleb ClarkMatt 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– It sometimes has a cycle creating a “chain
reaction”
Example Problems
Examples
• Find the Derivative of Sin(x )2
Examples
• Find the Derivative of Sin(x )2
Examples
• Find the Derivative of cos(x )3
Examples
• Find the Derivative of cos(x )3
Examples
• Find the Derivative of ln(x )2
Examples
• Find the Derivative of ln(x )2
Examples
• Find the Derivative of log (x )29
Examples
• Find the Derivative of log (x )29
Examples
• Find the Derivative of tan(x )4
Examples
• Find the Derivative of tan(x )4
Multiple Choice Questions
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
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
Multiple Choice Problem 2
• What is the derivative of ea. eb. 4ec. e ln4d. 4xe
4x
4x
4x
4x
4x
Multiple Choice Problem 2
• What is the derivative of ea. eb. 4ec. e ln4d. 4xe
4x
4x
4x
4x
4x
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
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
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)
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)
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
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
Free Response Question
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
Free Response
a. To the hundredth decimal place, how long does it take Pocahontas to reach John Smith?
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
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.
( )Free Response Solutions
a. Set x=- Te +sin(T) +50 equal to 0. 8
When Solved T= 84.57 minutes
tan (T)_______
-1
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
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
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