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

Section 2.8 The Derivative as a Function 153
The function f 9 is called the derivative of f because it has been “derived” from f by
the limiting operation in Equation 2. The domain of f 9 is the set hx | f 9sxd existsj and
may be smaller than the domain of f.
y
1
0
1
FIGURE 1
y=ƒ
x
ExamplE 1 The graph of a function f is given in Figure 1. Use it to sketch the graph
of the derivative f 9.
SOLUtion We can estimate the value of the derivative at any value of x by drawing the
tangent at the point sx, f sxdd and estimating its slope. For instance, for x − 5 we draw
the tangent at P in Figure 2(a) and estimate its slope to be about 3 2 , so f 9s5d < 1.5. This
allows us to plot the point P9s5, 1.5d on the graph of f 9 directly beneath P. (The slope
of the graph of f becomes the y-value on the graph of f 9.) Repeating this procedure at
several points, we get the graph shown in Figure 2(b). Notice that the tangents at A, B,
and C are horizontal, so the derivative is 0 there and the graph of f 9 crosses the x-axis
(where y − 0) at the points A9, B9, and C9, directly beneath A, B, and C. Between A and
B the tangents have positive slope, so f 9sxd is positive there. (The graph is above the
x-axis.) But between B and C the tangents have negative slope, so f 9sxd is negative
there.
y
B
1
m=0
A
m=0
y=ƒ
P
3
mÅ
2
0
1
m=0
5
x
TEC Visual 2.8 shows an animation
of Figure 2 for several functions.
(a)
C
y
1
y=fª(x)
Pª(5, 1.5)
0
Aª Bª Cª
1
5
x
FIGURE 2
(b)
n
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