Applied Calculus - 7e - c 03.pdf
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3
Differentiation
H
ow is a pond’s oxygen content
affected by organic waste? In
Example 7, page 178, you will
see how to find the rate at which
oxygen is being restored to the
pond after organic waste has been
dumped into it
.
T
HIS CHAPTER GIVES several rules that will greatly simplify the task of
finding the derivative of a function, thus enabling us to study how fast
one quantity is changing with respect to another in many real-world situ-
ations. For example, we will be able to find how fast the population of an
endangered species of whales grows after certain conservation measures
have been implemented, how fast an economy’s consumer price index (CPI)
is changing at any time, and how fast the time taken to learn the items on
a list changes with respect to the length of a list. We also see how these
rules of differentiation facilitate the study of marginal analysis, the study
of the rate of change of economic quantities. Finally, we introduce the
notion of the differential of a function. Using differentials is a relatively
easy way of approximating the change in one quantity due to a small
change in a related quantity.
159
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160
3
DIFFERENTIATION
3.1
Basic Rules of Differentiation
Four Basic Rules
The method used in Chapter 2 for computing the derivative of a function is based on
a faithful interpretation of the definition of the derivative as the limit of a quotient.
Thus, to find the rule for the derivative
f
of a function
f
, we first computed the dif-
ference quotient
f
1
x
h
2
f
1
x
2
h
and then evaluated its limit as
h
approached zero. As you have probably observed,
this method is tedious even for relatively simple functions.
The main purpose of this chapter is to derive certain rules that will simplify the
process of finding the derivative of a function. Throughout this book, we will use the
notation
d
dx
3
f
1
x
24
Read “
d
,
dx
of
f
of
x
”
to mean “the derivative of
f
with respect to
x
at
x
.” In stating the rules of differenti-
ation, we assume that the functions
f
and
g
are differentiable.
y
f
(
x
) =
c
Rule 1: Derivative of a Constant
d
dx
1
2
x
c
0
(
c
, a constant)
FIGURE
1
The slope of the tangent line to the graph
of
f
(
x
)
c
, where
c
is a constant, is zero.
The derivative of a constant function is equal to zero.
We can see this from a geometric viewpoint by recalling that the graph of a con-
stant function is a straight line parallel to the
x
-axis (Figure 1). Since the tangent line
to a straight line at any point on the line coincides with the straight line itself, its
slope [as given by the derivative of
f
(
x
)
c
] must be zero. We can also use the def-
inition of the derivative to prove this result by computing
f
1
x
h
2
f
1
x
2
f
¿
1
x
2
lim
h
S
0
h
c
c
lim
h
S
0
h
lim
h
S
0
0
0
EXAMPLE 1
a.
If
f
(
x
)
28, then
d
dx
f
¿
1
x
2
1
28
2
0
b.
If
f
(
x
)
2, then
d
dx
f
¿
1
x
2
1
2
2
0
03-W3979 11/2/06 12:42 PM Page 161
3.1
161
BASIC RULES OF DIFFERENTIATION
Rule 2: The Power Rule
d
dx
If
n
is any real number, then
1
x
n
2
nx
n
1
.
x
2
, then
Let’s verify the power rule for the special case
n
2. If
f
(
x
)
f
1
x
h
2
f
1
x
2
d
dx
x
2
f
¿
1
x
2
1
2
lim
h
S
0
h
1
x
h
2
2
x
2
lim
h
S
0
h
x
2
h
2
x
2
2
xh
lim
h
S
0
h
h
2
2
xh
lim
h
S
0
lim
h
S
0
1
2
x
h
2
2
x
h
as we set out to show.
The proof of the power rule for the general case is not easy to prove and will be
omitted. However, you will be asked to prove the rule for the special case
n
3 in
Exercise 77, page 170.
EXAMPLE 2
a.
If
f
(
x
)
x
, then
d
dx
1
#
x
1
1
x
0
f
¿
1
x
2
1
x
2
1
x
8
, then
b.
If
f
(
x
)
d
dx
x
8
8
x
7
f
¿
1
x
2
1
2
x
5/2
, then
c.
If
f
(
x
)
d
dx
5
2
x
3/2
x
5/2
f
¿
1
x
2
1
2
To differentiate a function whose rule involves a radical, we first rewrite the
rule using fractional powers. The resulting expression can then be differentiated
using the power rule.
EXAMPLE 3
Find the derivative of the following functions:
1
2
a.
f
1
x
2
1
x
b.
g
1
x
2
3
x
Solution
in the form
x
1/2
, we obtain
a.
Rewriting
1
x
d
dx
x
1/2
f
¿
1
x
2
1
2
1
2
x
1/2
1
2
x
1/2
1
2
1
x
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162
3
DIFFERENTIATION
1
2
in the form
x
1/3
, we obtain
b.
Rewriting
3
x
d
dx
x
1/3
g
¿
1
x
2
1
2
1
3
x
4/3
1
3
x
4/3
Rule 3: Derivative of a Constant Multiple of a Function
d
dx
c
d
dx
3
cf
1
x
24
3
f
1
x
24
(
c
, a constant)
The derivative of a constant times a differentiable function is equal to the constant
times the derivative of the function.
This result follows from the following computations.
If
g
(
x
)
cf
(
x
), then
g
1
x
h
2
g
1
x
2
cf
1
x
h
2
cf
1
x
2
g
¿
1
x
2
lim
h
S
0
lim
h
S
0
h
h
f
1
x
h
2
f
1
x
2
c
lim
h
S
0
h
cf
¿
1
x
2
EXAMPLE 4
a.
If
f
(
x
)
5
x
3
, then
d
dx
5
d
dx
5
x
3
x
3
f
¿
1
x
2
1
2
1
2
3
x
2
15
x
2
5
1
2
3
1
b.
If
f
1
x
2
, then
x
d
dx
3
x
1/2
f
¿
1
x
2
1
2
1
2
x
3/2
3
2
x
3/2
3
a
b
Rule 4: The Sum Rule
d
dx
d
dx
d
dx
3
f
1
x
2
g
1
x
24
3
f
1
x
24
3
g
1
x
24
The derivative of the sum (difference) of two differentiable functions is equal to the
sum (difference) of their derivatives.
This result may be extended to the sum and difference of any finite number of dif-
ferentiable functions. Let’s verify the rule for a sum of two functions.
03-W3979 11/2/06 12:42 PM Page 163
3.1
163
BASIC RULES OF DIFFERENTIATION
If
s
(
x
)
f
(
x
)
g
(
x
), then
s
1
x
h
2
s
1
x
2
s
¿
1
x
2
lim
h
S
0
h
3
f
1
x
h
2
g
1
x
h
24
3
f
1
x
2
g
1
x
24
lim
h
S
0
h
3
f
1
x
h
2
f
1
x
24
3
g
1
x
h
2
g
1
x
24
lim
h
S
0
h
f
1
x
h
2
f
1
x
2
g
1
x
h
2
g
1
x
2
lim
h
S
0
lim
h
S
0
h
h
f
¿
1
2
g
¿
1
2
x
x
EXAMPLE 5
Find the derivatives of the following functions:
t
2
5
5
t
3
4
x
5
3
x
4
8
x
2
1
2
a.
f
(
x
)
x
3
b.
g
t
Solution
d
dx
f
¿
1
2
1
4
x
5
3
x
4
8
x
2
2
a.
x
x
3
d
dx
d
dx
d
dx
d
dx
d
dx
4
x
5
3
x
4
8
x
2
1
2
1
2
1
2
1
x
2
1
3
2
20
x
4
12
x
3
16
x
1
b.
Here, the independent variable is
t
instead of
x
, so we differentiate with
respect to
t
. Thus,
d
dt
1
5
t
2
1
t
3
as
t
3
g
¿
1
2
¢
5
t
3
≤
t
Rewrite
.
2
5
t
15
t
4
2
t
5
75
t
4
as
1
Rewrite
t
4
and simplify.
5
t
4
EXAMPLE 6
Fi
n
d the slope and an equation of the tangent line to the graph of
at the point (1, 3).
1
2
1
f
x
2
x
1/
x
Solution
The slope of the tangent line at any point on the graph of
f
is given by
d
dx
1
1
f
¿
1
2
a
b
x
2
x
x
d
dx
1
1
1
x
1/2
x
1/2
1
2
x
2
x
1/2
Rewrite
x
as
.
1
2
x
3/2
2
Use the sum rule.
1
2
x
3/2
2
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