ACOW RATES OF CHANGE AND DERIVATIVES MODULE
Updated 5/28/2016
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Activity C2 – Product and Quotient Rules
When one function is defined as the product of two other functions, the product rule must
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be applied to find the derivative of the given function. The function h( x)  3x2  2 x  1 ,
for example, is the product of the function f ( x)  3x2 and g ( x)   2 x  1  4 x2  4 x  1 .
Thus, using the product rule, we can find h '( x) .
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h '( x)  f '( x) g ( x)  f ( x) g '( x)


 6 x 4 x 2  4 x  1  3x 2 8 x  4 
 24 x 3  24 x 2  6 x  24 x 3  12 x 2
 48 x 3  36 x 2  6 x
We can prove h '( x)  48x3  36 x2  6 x by rewriting h( x) in its expanded form.
h( x)  3x 2  2 x  1
2


 3x 2 4 x 2  4 x  1
 12 x  12 x  3 x
4
3
2
Applying the power rule gives h '( x)  48x3  36 x2  6 x . The following exercises will help
you discover an important concept regarding the product rule.
1.
a) Given f ( x)  3x2 find f '( x) .
b) Given g ( x)   2 x  1  4 x2  4 x  1 find g '( x ) .
c) Use the answers from 1a) and 1b) to find f '( x)  g '( x) .
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d) Compare the answer found in 1c) to the derivative of h( x)  3x2  2 x  1 found
above and then determine which of the following statements is a correct
conclusion?
a) h '( x)  g '( x)  f '( x)
b) h '( x)  f '( x)  g '( x)
c) h '( x)  g '( x)  f '( x)
d) No conclusions can be made.
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In the previous example it was possible to find the derivative of h( x) by rewriting h( x) in
its expanded form and then apply the power rule. This will not always be the case. For
some functions the product rule will have to be applied to find the derivative.
2. Which of the following is a correct way to find the derivative of y  7 x3 ( x2  2x  1) ?
a) y '  21x2 ( x2  2x  1)  7 x3 (2x  2)
b) y '  21x2 ( x2  2x  1)  7 x3 (2x  2)
ACOW RATES OF CHANGE AND DERIVATIVES MODULE
Updated 5/28/2016
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c) y '  21x2 (2x  2)
d) y '  7 x3 ( x2  2x  1)  7 x3 (2x  2)
3. If h( x)  f ( x)  6 x , f '(9)  4 , and f (9)  10 , find h '(9) .
4. Given the graphs of f ( x) and g ( x) (RS2), which of the following is true regarding h '(3)
if h( x)  f ( x) g ( x) ?
a) h '(3) is positive
b) h '(3) is negative
c) h '(3) is zero
d) h '(3) can not be found
1
x
5. If h( x)  f ( x) g ( x) , f ( x)  2 x3 , and g ( x)  5 x1/ 2  , find h '( x) .
6. If f ( x) 
1
2
 x  1 , find f '(2) .
x
If a given function is the quotient of two functions, then to take the derivative, we must
apply the quotient rule(RS3). The function below, for example, is the quotient of two
functions.
h( x ) 
x
3x  2
2
The quotient rule has to be applied to find the derivative of h(x)(RS4).
Some alternate forms of the quotient rule(RS5), have been formulated to make the
quotient rule easier for students to remember. It is important to notice the numerator of
the quotient rule contains subtraction and therefore the order in which the terms in the
numerator are written is important.
Find the derivatives of the following functions. Simplify your answer.
7. f ( x ) 
x
2x 1
8. g ( x) 
4x
6 x  3x 4
3
ACOW RATES OF CHANGE AND DERIVATIVES MODULE
Updated 5/28/2016
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Find the derivatives of the following functions. Do not simplify your answer.
9. h( x) 
10.
2 x
3
p( x) 
11. q( x) 
x 2  5 x 4
3x3/ 2  5 x 7
6 5 x3
2.1x3  x
2 x2  5