Inverse, Exponential, and Logarithmic Functions
Derivatives of e x and of the Natural Log
Derivatives of Exponential Functions
One truly remarkable characteristic of e x is that
e
x
 = e
x
|
Besides the trivial case of f (x) = 0 , e x and its constant multiples are the only functions whose derivatives are equal to themselves!
Incorporating the principles of the chain rule, we might also say that if u is a function of x , then
e
u = e
u
|
By extension,
e
x
dx = e
x + c
. Using the fact that
|
e
u = e
u
|
we can derive a more general formula for the derivative of a x , where a is any positive constant.
First, note that a x can be rewritten as
e ln(ax)
since e x and ln(x) are inverses of each other.
This quantity can also be written as
e x ln(a) .
Now,
a
x
 =
e
x ln(a)
|
By the chain rule,
a
x
=
e
x ln(a)
 ln(a)
|
Changing the exponential function back to its original form, we get
|
a
x
= a
x
ln(a)
|
By extension,
a
x
dx=
a
x+c
|
Derivatives of Logarithms
It may be satisfying to learn now that for x >0,
ln(x) = 
|
The appeal rests in the corresponding implication that
=ln
x
 +c
|
Recall that the power rule did not offer a way of integrating the function
,
but now it is possible to do so.
A related rule for logarithms of any base is that
loga(x) = 
|
Logarithmic Differentiation
To find the derivative of a constant raised to a power of x , the rule presented earlier in this section should suffice. However, to find the derivative of a function of x that is raised to a power of x , the technique of logarithmic differentiation is necessary.
Example: Differentiate
y = x
3x
.
Step One: Take the natural log of both sides of the equation:
ln(y) = ln(x
3x)
.
Step Two: Now use log rules to take the variable
x
out of the exponent and
turn it into a product:
ln(y) = (3x)(ln(x))
.
Step Three: Implicitly differentiate both sides with respect to
x
(remember to use the
chain rule):
 = 3x
 +3 ln(x)
|
Step Four: Solve for
algebraically:
|
|
=  3+3 ln(x)
y
|
||
|
|
= 3+3 ln(x)
x
3x
|
||
|
|
= 3x 3x +3x 3xln(x) |
