James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

172 Chapter 3 Differentiation Rules
y
c
y=c
slope=0
In this section we learn how to differentiate constant functions, power functions, polynomials,
and exponential functions.
Let’s start with the simplest of all functions, the constant function f sxd − c. The graph
of this function is the horizontal line y − c, which has slope 0, so we must have f 9sxd − 0.
(See Figure 1.) A formal proof, from the definition of a derivative, is also easy:
f 9sxd − lim
h l 0
f sx 1 hd 2 f sxd
h
c 2 c
− lim − lim 0 − 0
h l 0 h h l 0
0 x
FIGURE 1
The graph of f sxd − c is the line
y − c, so f 9sxd − 0.
In Leibniz notation, we write this rule as follows.
Derivative of a Constant Function
d
dx scd − 0
y
y=x
Power Functions
We next look at the functions f sxd − x n , where n is a positive integer. If n − 1, the
graph of f sxd − x is the line y − x, which has slope 1. (See Figure 2.) So
0
slope=1
FIGURE 2
The graph of f sxd − x is the line
y − x, so f 9sxd − 1.
x
1
d
dx sxd − 1
(You can also verify Equation 1 from the definition of a derivative.) We have already
investigated the cases n − 2 and n − 3. In fact, in Section 2.8 (Exercises 19 and 20) we
found that
2
d
dx sx 2 d − 2x
d
dx sx 3 d − 3x 2
For n − 4 we find the derivative of f sxd − x 4 as follows:
f 9sxd − lim
h l 0
f sx 1 hd 2 f sxd
h
− lim
h l 0
sx 1 hd 4 2 x 4
h
− lim
h l 0
x 4 1 4x 3 h 1 6x 2 h 2 1 4xh 3 1 h 4 2 x 4
h
− lim
h l 0
4x 3 h 1 6x 2 h 2 1 4xh 3 1 h 4
h
− lim
h l 0
s4x 3 1 6x 2 h 1 4xh 2 1 h 3 d − 4x 3
Thus
3
d
dx sx 4 d − 4x 3
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