James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 3.6 Derivatives of Logarithmic Functions 221
If we hadn’t used logarithmic differentiation
in Example 7, we would have
had to use both the Quotient Rule
and the Product Rule. The resulting
calculation would have been
horrendous.
Because we have an explicit expression for y, we can substitute and write
dy
dx − x 3y4 sx 2 1 1S 3
s3x 1 2d 5 4x 1 x
x 2 1 1 2 15
3x 1 2D
■
Steps in Logarithmic Differentiation
1. Take natural logarithms of both sides of an equation y − f sxd and use the Laws
of Logarithms to simplify.
2. Differentiate implicitly with respect to x.
3. Solve the resulting equation for y9.
If f sxd , 0 for some values of x, then ln f sxd is not defined, but we can write
| y | − | f sxd | and use Equation 4. We illustrate this procedure by proving the general
version of the Power Rule, as promised in Section 3.1.
The Power Rule If n is any real number and f sxd − x n , then
f 9sxd − nx n21
If x − 0, we can show that f 9s0d − 0
for n . 1 directly from the definition
of a derivative.
Proof Let y − x n and use logarithmic differentiation:
Therefore
ln | y | − ln | x | n − n ln | x | x ± 0
y9
y − n x
Hence
y9 − n y x − n x n
x
− nx
n21
■
Constant base, constant exponent
Variable base, constant exponent
Constant base, variable exponent
Variable base, variable exponent
You should distinguish carefully between the Power Rule fsx n d9 − nx n21 g, where the
base is variable and the exponent is constant, and the rule for differentiating exponential
functions fsb x d9 − b x ln bg, where the base is constant and the exponent is variable.
In general there are four cases for exponents and bases:
d
1.
dx sb n d − 0 (b and n are constants)
d
2.
dx f f sxdgn − nf f sxdg n21 f 9sxd
d
3.
dx fb tsxd g − b tsxd sln bdt9sxd
4. To find sdydxdf f sxdg tsxd , logaritmic differentiation can be used, as in the next
example.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.