The Chain Rule

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