Algorithms

compute their derivatives and integrals. We also define the general logarithmic functions
such as and and find their derivatives and integrals as well.
The Derivative of
We start with the definition
If then
With the Chain Rule, we get a more general form.
d
dx ax = ax ln a.
a 7 0,
= ax ln a.
d
dx
eu = eu
du
dx
d
dx ax = d
dx ex ln a = ex ln a # d
dx sx ln ad
ax = ex ln a :
au
log2 x, log10 x, logp x,
p2x, 10x , x .
7.4 ax loga x
If and u is a differentiable function of x, then is a differentiable function
of x and
(1)
d
dx au = au ln a
du
dx
.
a 7 0 au
These equations show why is the exponential function preferred in calculus. If
then and the derivative of simplifies to
d
dx ex = ex ln e = ex .
ln a = 1 ax
ex a = e,
EXAMPLE 1 Differentiating General Exponential Functions
(a)
(b)
(c)
From Equation (1), we see that the derivative of is positive if or
and negative if or Thus, is an increasing function of x if
and a decreasing function of x if In each case, is one-to-one. The second
derivative
is positive for all x, so the graph of is concave up on every interval of the real line
(Figure 7.12).
Other Power Functions
The ability to raise positive numbers to arbitrary real powers makes it possible to define
functions like and for We find the derivatives of such functions by rewriting
the functions as powers of e.
EXAMPLE 2 Differentiating a General Power Function
Find
Solution Write as a power of e:
Then differentiate as usual:
The Integral of
If so that we can divide both sides of Equation (1) by ln a to obtain
au
du
dx = 1
ln a
d
dx saud.
a Z 1, ln a Z 0,
au
= xx s1 + ln xd.
= xx ax # 1x
+ ln xb
= ex ln x
d
dx sx ln xd
dy
dx = d
dx ex ln x
y = xx = ex ln x . ax with a = x.
xx
dy>dx if y = xx, x 7 0.
xx xln x x 7 0.
ax
d2
dx2 saxd = d
dx sax ln ad = sln ad2 ax
0 6 a 6 1. ax
ln a 6 0, 0 6 a 6 1. ax a 7 1
ax ln a 7 0, a 7 1,
d
dx 3sin x = 3sin x sln 3d
d
dx ssin xd = 3sin x sln 3d cos x
d
dx 3-x = 3-x sln 3d
d
dx s -xd = -3-x ln 3
d
dx 3x = 3x ln 3
496 Chapter 7: Transcendental Functions
x
y
–1 0 1
1
x
y
1x
y