Derivatives of Exponential Functions

One truly remarkable characteristic of e ^x is that

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

By extension, e ^x dx = e ^x + c . Using the fact that

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,

By the chain rule,

Changing the exponential function back to its original form, we get

By extension,

Derivatives of Logarithms

It may be satisfying to learn now that for x >0,

The appeal rests in the corresponding implication that

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

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):

Step Four: Solve for algebraically: