a. Use the Product Rule to find the derivative of the given
function.
b. Find the derivative by expanding the product first.
f(x) = (x − 7)(2x + 4)
a. Use the product rule to find the derivative of the function.
Select the correct answer below and fill in the answer box(es)
to complete your choice.
A. The derivative is (x − 7)(2x + 4) .
B. The derivative is (x − 7)(2x + 4) + .
C. The derivative is (x − 7).
D. The derivative is x(2x + 4).
E. The derivative is (x − 7) + (2x + 4)
b. Expand the product.
(x − 7)(2x + 4) = (Simplify your answer.)
Using either approach, .
d
dx
(x − 7)(2x + 4) =

1. a) Use the Product Rule to find the derivative of the given
   function.
   b) Find the derivative by multiplying the expressions first.
   y = 9 x + 4 x
   2
   a) Use the Product Rule to find the derivative of the function.
   Select the correct answer below and fill in the answer box(es)
   to complete your choice.
   A. The derivative is x +
   2
   B. The derivative is 9 x + 4 x + .
   2
   C. The derivative is 9 x + 4 + x
   2
   D. The derivative is 9 x + 4 .
   b) Multiply the expressions.

9 x + 4 x (Simplify your answer.) 2

Now take the derivative of the answer from the previous step
and simplify the answer from part a. Check to make sure that
the two results are the same. That is, using either approach,
y′ = .

1. a) Use the Quotient Rule to find the derivative of the given
   function.
   b) Find the derivative by dividing the expressions first.
   y = for x 0
   x
   5
   x
   3
   ≠
   a) Use the Quotient Rule to find the derivative of the given
   function. Select the correct answer below and fill in the
   answer box(es) to complete your choice.
   A. The derivative is
   x
   5
   • + x
   3
   x
   6
   B. The derivative is
   x
   3
   • − x
   5
   x
   6
   C. The derivative is
   x
   3
   • − x
   5
   x
   5
   D. The derivative is
   x
   5
   • − x
   3
   x
   5
   b) Divide the expressions.
   (Simplify your answer.)
   x
   5
   x

3

Now take the derivative of the answer from the previous step
and simplify the answer from part a. Check to make sure that
the two results are the same. That is, using either approach,
.
dy

dx

1. Use the quotient rule to find the derivative of the given
   function. Then find the derivative by first simplifying the
   function. Are the results the same?
   h(w) =
   4w
   5
   − w
   w
   What is the immediate result of applying the quotient rule?
   Select the correct answer below.
   A. 4w − 1
   4
   B. w 20w
   4
   − 1 − 4w
   5
   − w (1)
   w
   2
   C. 16w
   3
   D. 20w − 1 (w) + 4w − w (1)
   4 5
   What is the fully simplified result of applying the quotient rule?
   What is the result of first simplifying the function, then taking
   the derivative? Select the correct answer below.
   A. 16w
   3
   B. 20w − 1 (w) + 4w − w (1)
   4 5
   C. 4w − 1
   4
   5.
   6.
   7.
   D. w 20w
   4
   − 1 − 4w
   5
   − w (1)
   Are the two results the same?
   No
   Yes
   Differentiate the function.
   y = 2x − x + 1 − x + 8
   4 5
   y′ =
   Differentiate the function.
   y =
   3x
   2
   − 4
   2x
   3

• 7
  y

′

Differentiate.
F(x) = (2x + 3)
2
F′(x) =
8.
9.
10.
Differentiate.
g(x) = x
6x − 5
7x + 3
+
3
g′(x) =
Use the quotient rule to find the derivative of the following.
y =
x
2
− 5x + 1
x
2

• 7
  y

′

Find an equation of the tangent line to the graph of
y = at the origin and at the point
2x
x
2

• 1
  (1,1).
  The tangent to the curve at the origin is y = .
  The tangent to the curve at the point ( , ) is
  y .

1 1

(Simplify your answer.)
11.
(1) increasing
decreasing
The gross domestic product (in billions of dollars) can be
approximated by P(t) , where t is the
number of years since 1960.
= 569 + t 36t − 107
0.6
a) Find P′(t).
b) Find P′(45).
c) In words, explain what P′(45) represents.
a) Find P′(t).
P′(t) =
b) Find P′(45).
P′(45) =
(Simplify your answer. Type an integer or decimal rounded to
the nearest tenth as needed.)
c) In words, explain what P′(45) represents.
In the year , the gross domestic product was
(1) by about billion dollars
per year.
(Type integers or decimals.)

1. The temperature T of a person
   during an illness is given by
   , where T is the
   temperature, in degrees
   Fahrenheit, at time t, in hours.
   T(t) = + 98.6
   7t
   t
   2

• 4
  (a) Find the rate of change of the
  temperature with respect to time.
  (b) Find the temperature at t = 1 hr.
  (c) Find the rate of change of the
  temperature at t = 1 hr.
  0 10
  98
  104
  t
  T(t)
  (a) Find the rate of change of the temperature with respect to
  time.
  T′(t) =
  (b) Find the temperature at t = 1 hr.
  T(1) =
  °F
  (Round to two decimal places as needed.)
  (c) Find the rate of change of the temperature at t = 1 hr.
  T′(1) =
  °F / hr
  (Round to two decimal places as needed.)

Sample Solution