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Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

198 Chapter 2 • The Derivative and Its Properties

x
Suppose f (x) = a , where a > 0 and a 6= 1. The derivative of f is
h
x
f (x + h) − f (x) a x + h − a x a · a − a x
′
f (x) = lim = lim = lim
h→0 h h→0 h ↑ h→0 h
x
a x + h = a · a h
a − 1 a − 1
h h
x x
= lim a · = a · lim
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h→0 h h→0 h
↑
x
Factor out a .
h
a − 1
provided lim exists.
h→0 h
x
Three observations about the derivative of f (x) = a are significant:
h
h
a − 1 a − 1
0
• f (0) = a lim = lim .
′
h→0 h h→0 h
d
x
x
x
′
• f (x) is a multiple of a . In fact, a = f (0) · a .
′
dx
• If f (0) exists, then f (x) exists, and the domain of f is the same as that of
′
′
′
x
f (x) = a , all real numbers.
The slope of the tangent line to the graph of f (x) = a x at the point (0, 1)
h
a − 1
′
is f (0) = lim , and the value of this limit depends on the base a. In Section P.5,
NEED TO REVIEW? The number e is
h→0 h
discussed in Section P.5, pp. 46–47.
the number e was defined as that number for which the slope of the tangent line to the
x
x
graph of y = a at the point (0, 1) equals 1. That is, if f (x) = e , then f (0) = 1 so that
′
y
f(x) e x e − 1
h
4 lim = 1
y x ! 1 h→0 h
Figure 28 shows f (x) = e x and the tangent line y = x + 1 with slope 1 at the
2 point (0, 1).
d x x d x x x x
x
′
Since a = f (0) · a , if f (x) = e , then e = f (0) · e = 1 · e = e .
′
(0, 1)
dx dx
2 2 4 x
THEOREM Derivative of the Exponential Function y = e x
Figure 28 The derivative of the exponential function y = e is
x
d
′ x x
y = e = e (1)
dx
EXAMPLE 7 Differentiating an Expression Involving y = e x
CALC CLIP
x
3
Find the derivative of f (x) = 4e + x .
Solution
3
x
The function f is the sum of 4e and x . Then
d d d d
′ x 3 x 3 x 2 x 2
f (x) = (4e + x ) = (4e ) + x = 4 e + 3x = 4e + 3x
dx ↑ dx dx ↑ dx ↑
Sum Rule Constant Multiple Rule; Use (1).
Simple Power Rule
R
NOW WORK Problem 25 and AP Practice Problem 4.
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