Page 170 - Calculus Demystified

CHAPTER 6
and Transcendental Functions 157

du
exp(u) dx = exp(u) + C.
dx
We note for the record that the exponential function is the only function (up to
constant multiples) that is its own derivative. This fact will come up later in our
applications of the exponential
EXAMPLE 6.9
Compute the derivatives:
d d d
exp(4x), (exp(cos x)), ([exp(x)]·[cot x]).
dx dx dx
SOLUTION
For the first problem, notice that u = 4x hence du/dx = 4. Therefore we
have
d d
exp(4x) =[exp(4x)]· (4x) = 4 · exp(4x).
dx dx
Similarly,

d d
(exp(cosx))=[exp(cosx)]· cosx =[exp(cosx)]·(−sinx),
dx dx

d d d
([exp(x)]·[cotx])= exp(x) ·(cotx)+[exp(x)]· cotx
dx dx dx
2
=[exp(x)]·(cotx)+[exp(x)]·(−csc x).
You Try It: Calculate (d/dx)(exp(x · sin x)).
EXAMPLE 6.10
Calculate the integrals:

3
exp(5x) dx, [exp(x)] dx, exp(2x + 7)dx.
SOLUTION
We have

1
exp(5x) dx = exp(5x) + C
5

3
[exp(x)] dx = [exp(x)]·[exp(x)]·[exp(x)] dx

1
= exp(3x) dx = exp(3x) + C
3

1 1
exp(2x + 7)dx = exp(2x + 7) · 2 dx = exp(2x + 7) + C.
2 2