The Chain Rule The final rule of taking derivatives that we need to learn (although it won’t seem like it’s the last rule since you’ll find yourself doing all sorts of weird problems in weird ways…) is called the Chain Rule. We’re going to prove it but you don’t really gain much in the way of insight by going through the proof. We’re just doing it because, you…I mean, people climb mountains and stuff like that, right?

f ( g ( x + h )) − f ( g ( x )) dy = lim . Let’s do it! Oh, I should dx h→0 h mention that we’re going to do two weird things at two key points. Here they are ahead of time: g ( x + h) − g ( x) • Multiply what we have by . g ( x + h) − g ( x) • Let k = g ( x + h ) − g ( x ) . So, given y = f ( g ( x )) , we need to find

There are several ways to write the chain rule. Here are a few of them: d d ⎡⎣ f ( g ( x )) ⎤⎦ = f ′ ( g ( x )) ⋅ g′ ( x ) • • ⎡ f ( u ) ⎤⎦ = f ′ ( u ) ⋅ du dx dx ⎣ dy dy du dy dy du dw dz • • = ⋅ = ⋅ ⋅ ⋅ , etc. dx du dx dx du dw dz dx That last one can just keep going forever as long as the functions are related. The “f-u” form is my favorite…because I’m immature. It might not be obvious at first, but you can use the units involved in a problem to work out what type of chain rule to throw at it. You’ll actually find yourself doing that a lot when a problem becomes confusing.

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For reasons that aren’t totally obvious to me, the chain rule seems to make the most sense to the most people when they use it on trig functions…so…let’s start there! Problem: Find the first derivative of each of the following functions. a. y = cos ( 4x ) b. y = sec x 4

( )

d. y = cot ( 8x )

c. y = sin 2 ( x )

e. y = csc ( sin ( x ))

f.

y = tan

(

x +2

)

The next most common place where the chain rule starts to really click is taking derivatives of ridiculous powers of polynomials. So…obviously… Problem: Find the first derivative of each of the following functions. 7 5 a. y = ( 3x − 2 ) b. y = (19 − 3x 4 )

c. y = ( x 2 + 3x + 2 )

4

We’ll often use the chain rule to find higher order derivatives. In calculus the best idea is usually not to expand! Problem: Find the second derivative of each of the following. 2 3 a. y = ( 3x + 5 ) b. y = ( 8 − 5x 2 )

Problem: Find the critical points of the function y = ( 2x − 3) ( 3x + 4 ) . 3

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Problem: Quickly and efficiently find the critical points of the given functions. Treat this as a race—a race against history. Someone will be or has been the fastest person to ever finish these problems… 2 3 5 3 a. y = ( 9x − 2 ) ( 4x + 5 ) b. y = ( x − 5 ) ( 2x + 7 )

c. y = ( 3x − 5 ) ( 2x + 3) 3

d. y = ( 5 − 3x ) ( 4x + 1)

4

3

5

Lots of problems will not involve given functions, but rather will just involve function notation. Then you’ll be given specific values and you have to figure out what to do with those. Here are some problems to warm you up to that idea. Problem: Find the first derivative of each of the following. a. y = f g ( x 2 ) b. y =

(

)

3

f ( sin ( x ))

Problem: Given that h ( x ) = ⎡⎣ f ( g ( x )) ⎤⎦ and the table of values below. 2

a. Find h ( 2 ) and h′ ( 2 ) .

x 2 5

f ( x) 4 -6

f ′( x) 3 -4

g( x) 5 2

g′ ( x ) -3 7

b. Approximate h ( 2.1) using the tangent line to h ( x ) at x = 2 .

c. Approximate h′ ( 3.4 ) using the data in the table.

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Problem: If f ( x ) = ( 4x 2 + 3x ) and g ( x ) = 4x − 5 , find the derivative of f ( g ( x )) at x = 1 . Do this 5

problem without preforming the composition! So much easier!

Problem: The graphs of f and g are shown below. Use them to find the derivative of each function at the given value. Graph of f ( x ) Graph of g ( x )

a.

f ( g ( x )) at x = −3

c.

f g ( x 2 ) at x = 1.5

d. ⎡⎣ f ( x 3) ⎤⎦ at x = −3

e.

f ( 2x ) ⋅ g ( 4x ) at x = 1

f.

(

)

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b. g ( f ( 2x )) at x = 2.5

2

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⎡⎣ g ( sin ( 2x )) ⎤⎦ at x = π 2 3

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Problem: Find the first derivative of each of the following functions. 2 b. y = ln 5x 2 + 6 a. y = e5t +sin(t )

(

c. y = log 3 ( sin ( 3x ) + 2 )

d. y = 6 9x + 2

e. y = ( 7x + 1) ( 3x − 2 )

f.

2

g. y = (1− x 2 )

11

3

)

y = e tan ( 3x ) 2

h. y = ( 9 − 5x )

( (

7

))

Problem: Find the first derivative of the function y = sin cos sec 2 ( tan ( 2x )) .

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The next question is very, very similar to problems that have appeared on the AP exam in (kind of) recent years.

(

Problem: For time t ≥ 0 hours, let r ( t ) = 80 1− e−15t

2

) represent the speed, in kilometers per hour, at

which a car travels along a straight road. The number of liters of gasoline used by the car to travel x kilometers is modeled by g ( x ) = 0.07x 1− e− x 3 .

(

)

a. At what time does the car’s speed reach 65 kilometers per hour?

b. What is the average rate of change of the car’s speed on the time interval 0 ≤ t ≤ 5 hours? Indicate units of measure.

c. How much gasoline does the car use to travel 1000 kilometers?

d. Write down the units for each of the functions r ( t ) , r ′ ( t ) , g ( x ) , and g′ ( x ) . (It might help to sketch axes for each function.)

e. Find the rate of change with respect to time of the number of liters of gasoline used by the car when t = 3 hours. Indicate units of measure. (One part of this is tricky…you’ll need to think back to Math Analysis. When you figure out how to do it use 10,000 as the upper bound.)

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