文档介绍：Section
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The Exponential Function
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At this point we have seen all the major concepts of calculus: derivatives, integrals, and
power series. For the rest of the book we will be concerned with how these ideas apply
in various circumstances. In particular, in this chapter we will introduce the remaining
elementary functions of calculus: the exponential function, the natural logarithm function,
the inverse trigonometric functions, and the hyperbolic trigonometric functions. As they
are introduced, we will discuss related issues involving derivatives, integrals, and power
series, as well as applications to the physical world.
We will begin by considering the exponential function. We first saw this function in
Section , but we will redefine it here pleteness.
Definition The exponential function, with value at x denoted by exp(x), is defined by
∞
xn x2 x3
exp(x) = = 1 + x + + + · · · . ()
X n! 2! 3!
n=0
We saw in Section that this series converges absolutely for all values of x; hence
the domain of the exponential function is (−∞, ∞) . We should also note that exp(0) = 1.
Using the properties of power series, it is an easy matter pute the derivative of
the exponential function:
∞∞∞∞
d d xn d xn xn−1 xn
exp(x) = ! = = = = exp(x).
dx dx X n! X dx n! X (n − 1)! X n!
n=0 n=0 n=1 n=0
Proposition
d
exp(x) = exp(x). ()
dx
Example Using the chain rule, we have
d
exp(4x) = 4 exp(x).
dx
Example Similarly,
d
exp(x2) = 2x exp(x2).
dx
In fact, the exponential function is the only function f for which both f(0) = 1 and
f 0(x) = f(x) for all x. To see this, we first demonstrate a more general property. Suppose
1
2 The Exponential Function Section
f is any function for which f(0) = c and f 0(x) = kf(x) for all x, where c and k are
constants. Then it follows that
d
f 00(x) = (kf(x)) = kf 0(x) = k2f(x),
dx
d