5128_Ch03_pp098-184

172 Chapter 3 Derivatives
3.9
What you’ll learn about
• Derivative of e x
• Derivative of a x
• Derivative of lnx
• Derivative of log a x
• Power Rule for Arbitrary Real
Powers
. . . and why
The relationship between
exponential and logarithmic
functions provides a powerful
differentiation tool called
logarithmic differentiation.
X
–.03
–.02
–.01
0
.01
.02
.03
X=0
[–4.9, 4.9] by [–2.9, 2.9]
(a)
Y1
.98515
.99007
.99502
ERROR
1.005
1.0101
1.0152
(b)
Figure 3.56 (a) The graph and (b) the
table support the conclusion that
1
lim
h→0 eh 1.
h
Derivatives of Exponential and
Logarithmic Functions
Derivative of e x
At the end of the brief review of exponential functions in Section 1.3, we mentioned that
the function y e x was a particularly important function for modeling exponential
growth. The number e was defined in that section to be the limit of 1 1x x as x→.
This intriguing number shows up in other interesting limits as well, but the one with the
most interesting implications for the calculus of exponential functions is this one:
1
lim
h→0 eh 1.
h
(The graph and the table in Figure 3.56 provide strong support for this limit being 1. A
formal algebraic proof that begins with our limit definition of e would require some rather
subtle limit arguments, so we will not include one here.)
The fact that the limit is 1 creates a remarkable relationship between the function e x
and its derivative, as we will now see.
d
e d x
x lim e xh e
x
h→0 h
h e x
lim e x •
h→0
eh
lim
h→0 ( e x • eh 1
h
e x • lim
h→0 (
e x • 1
e x
)
)
1
eh h
In other words, the derivative of this particular function is itself!
d
e d x
x e x
If u is a differentiable function of x, then we have
d
e d x
u e u d u
.
dx
We will make extensive use of this formula when we study exponential growth and
decay in Chapter 6.
EXAMPLE 1 Using the Formula
Find dydx if y e (xx2) .
SOLUTION
Let u x x 2 then y e u . Then
dy
e u du
du
, and 1 2x.
dx dx
dx
dy
Thus, e u du
e (xx2) (1 2x).
dx dx
Now try Exercise 9.