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Calculus volume 1 Derivatives Derivatives of exponential and
Problem-solving strategy: using logarithmic differentiation
To differentiate
y
=
h
(
x
) using logarithmic differentiation, take the natural logarithm of both sides of the equation to obtain
ln
y
=
ln
(
h
(
x
)
)
.
Use properties of logarithms to expand
ln
(
h
(
x
)
) as much as possible.
Differentiate both sides of the equation. On the left we will have
1
y
d
y
d
x
.
Multiply both sides of the equation by
y to solve for
d
y
d
x
.
Replace
y by
h
(
x
)
.
Using logarithmic differentiation
Find the derivative of
y
=
(
2
x
4
+
1
)
tan
x
.
Use logarithmic differentiation to find this derivative.
ln
y
=
ln
(
2
x
4
+
1
)
tan
x
Step 1. Take the natural logarithm of both sides.
ln
y
=
tan
x
ln
(
2
x
4
+
1
)
Step 2. Expand using properties of logarithms.
1
y
d
y
d
x
=
sec
2
x
ln
(
2
x
4
+
1
)
+
8
x
3
2
x
4
+
1
·
tan
x
Step 3. Differentiate both sides. Use the
product rule on the right.
d
y
d
x
=
y
·
(
sec
2
x
ln
(
2
x
4
+
1
)
+
8
x
3
2
x
4
+
1
·
tan
x
)
Step 4. Multiply by
y
on both sides.
d
y
d
x
=
(
2
x
4
+
1
)
tan
x
(
sec
2
x
ln
(
2
x
4
+
1
)
+
8
x
3
2
x
4
+
1
·
tan
x
)
Step 5. Substitute
y
=
(
2
x
4
+
1
)
tan
x
.
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Using logarithmic differentiation
Find the derivative of
y
=
x
2
x
+
1
e
x
sin
3
x
.
This problem really makes use of the properties of logarithms and the differentiation rules given in this chapter.
ln
y
=
ln
x
2
x
+
1
e
x
sin
3
x
Step 1. Take the natural logarithm of both sides.
ln
y
=
ln
x
+
1
2
ln
(
2
x
+
1
)
−
x
ln
e
−
3
ln
sin
x
Step 2. Expand using properties of logarithms.
1
y
d
y
d
x
=
1
x
+
1
2
x
+
1
−
1
−
3
cos
x
sin
x
Step 3. Differentiate both sides.
d
y
d
x
=
y
(
1
x
+
1
2
x
+
1
−
1
−
3
cot
x
)
Step 4. Multiply by
y
on both sides.
d
y
d
x
=
x
2
x
+
1
e
x
sin
3
x
(
1
x
+
1
2
x
+
1
−
1
−
3
cot
x
)
Step 5. Substitute
y
=
x
2
x
+
1
e
x
sin
3
x
.
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Extending the power rule
Find the derivative of
y
=
x
r where
r is an arbitrary real number.
The process is the same as in
[link] , though with fewer complications.
ln
y
=
ln
x
r
Step 1. Take the natural logarithm of both sides.
ln
y
=
r
ln
x
Step 2. Expand using properties of logarithms.
1
y
d
y
d
x
=
r
1
x
Step 3. Differentiate both sides.
d
y
d
x
=
y
r
x
Step 4. Multiply by
y
on both sides.
d
y
d
x
=
x
r
r
x
Step 5. Substitute
y
=
x
r
.
d
y
d
x
=
r
x
r
−
1
Simplify.
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Key concepts
On the basis of the assumption that the exponential function
y
=
b
x
,
b
>
0 is continuous everywhere and differentiable at 0, this function is differentiable everywhere and there is a formula for its derivative.
We can use a formula to find the derivative of
y
=
ln
x
, and the relationship
log
b
x
=
ln
x
ln
b allows us to extend our differentiation formulas to include logarithms with arbitrary bases.
Logarithmic differentiation allows us to differentiate functions of the form
y
=
g
(
x
)
f
(
x
) or very complex functions by taking the natural logarithm of both sides and exploiting the properties of logarithms before differentiating.
Key equations
Derivative of the natural exponential function
d
d
x
(
e
g
(
x
)
)
=
e
g
(
x
)
g
′
(
x
)
Derivative of the natural logarithmic function
d
d
x
(
ln
g
(
x
)
)
=
1
g
(
x
)
g
′
(
x
)
Derivative of the general exponential function
d
d
x
(
b
g
(
x
)
)
=
b
g
(
x
)
g
′
(
x
)
ln
b
Derivative of the general logarithmic function
d
d
x
(
log
b
g
(
x
)
)
=
g
′
(
x
)
g
(
x
)
ln
b
For the following exercises, find
f
′
(
x
) for each function.
Source:
OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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