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DEOET
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2002 Doug MacLean
Derivatives of Logarithms
Recall that the logarithm functions satisfy very important arithmetic laws:If
a
and
b
are positive numbers, and
c
is a positive number not equal to 1, we have:
log
c
a
ln
a
ln
c
ln
(ab)
ln
a
ln
b
log
c
(ab)
log
c
a
log
c
b
ln
a b
ln
a
ln
b
log
c
a b
log
c
a
log
c
b
ln
a
b
b
ln
a
log
c
a
b
b
log
c
a
The natural logarithm function was defined to the inverse of the exponential function
e
x
, and the base
a
logarithms were defined to
be the inverse of the exponential function
a
x
, so we have the Cancellation Laws
e
ln
x
x
and
a
ln
x
x
so if we let
y
ln
x
or
y
log
a
x
we have
e
y
x
or
a
y
x
Using the method of implicit differentiation, we get:
(e
y
′
e
y
y
′
(x)
′
or
(a
y
′
ln
a)a
y
y
′
(x)
′
so
y
′
e
y
(^1) x
or
y
′
ln
a)a
y
ln
a)x
Thus
d dx
ln
x)
(^1) x
and
d dx
log
a
x
ln
a)x
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a
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2002 Doug MacLean
We may get more general formulas by using the Chain Rule:
d dx
ln
f (x))
f
′
(x)
f (x)
and
d dx
log
a
f (x)
f
′
(x)
ln
a)f (x)
ln
a
f
′
(x)
f (x)
The quantity
f
′
(x)
f (x)
is called the
relative rate of change
of the function
f
, and is very important in practical applications.
Example 1:
Find the derivative of
y
ln
(x
5
x
3
Solution:
y
′
x
5
x
3
(x
5
x
3
′
x
5
x
3
x
4
x
2
x
2
x
2
x
5
x
3
Example 2:
Find
d dx
sin
ln
(x))
Solution: d dx
sin
ln
(x))
cos
ln
x)
(^1) x
cos
ln
x)
x
U
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a n e n s
DEOET
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2002 Doug MacLean
Logarithmic Differentiation
One of the most important uses of the natural logarithm function is in the computation of derivatives of functions which are madeup of products, quotients and powers of more elementary functions. We use the three basic arithmetic properties of the logarithm tosimplify the function.
Example 5:
Find
y
′
if
y
(x
4
(x
3
(x
7
Solution:
Taking logarithms of both sides of the equation, we get
ln
y
ln
(x
4
(x
3
(x
7
or
ln
y
4 ln
(x
3 ln
(x
7 ln
(x
which we now differentiate:^ y
′
y
x
x
x
which we need only simplify slightly to get
y
′
in a usable form:
y
′
y
[
x
x
x
]
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tc
h
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DEOET
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2002 Doug MacLean
Example 6:
Find
y
′
if
y
x
4
π
x
sin
5
x)
(x
3
(x
7
. 5
Solution:
Taking logarithms of both sides of the equation, we get
ln
y
ln
x
4
π
x
sin
5
x)
(x
3
(x
7
. 5
or
ln
y
4 ln
x
x
ln
π
5 ln sin
x)
3 ln
(x
5 ln
(x
which we now differentiate:^ y
′
y
1 x
ln
π
3 cos
x)
sin
x)
x
x
which we need only simplify slightly to get
y
′
in a usable form:
y
′
y
[
(^4) x
ln
π
15 cot
x)
x
x
]
U
n iv
e r s it a s
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tc
h
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a n e n s
DEOET
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2002 Doug MacLean
Negative
x
There will be occasions when we wish to apply logarithms and deal with negative values of the variables concerned.Of course, ln
x
is undefined if
x
- However, ln
x
is
defined if
x <
Let us then find the derivative of ln
x
for non-zero
x
If
x >
0, it is of course
1 x
If
x <
0, then
x
x
, so ln
x
ln
x)
, and we can apply the Chain Rule:
d dx
ln
x))
x
d dx
x)
x
(^1) x
so we have the important formula
d dx
ln
x
(^1) x
if
x
