# Derivatives Of Exponential Functions

Exponential Functions: A Brief Review

An exponential function is a function of the form:

where is a non-zero number and is a variable.

One should be careful and make the distinction between an exponential function such as versus a polynomial such as which is a variable to a numeric power/exponent.

The Derivative of An Exponential Function

If we are given where is a different function of . The derivative of is:

Note that this general formula does use a variation of the chain rule. Since the exponent is a function of x, we take the derivative of the exponent as well.

Given the more common case of in the exponent, the general case becomes:

If we are given as the base such that we have . The derivative will be as follows:

(Note that ln(e) = 1 as .)

A special case is where

In this case we have and . The derivative of is simply .

Examples

Example One:

Example Two:

Example Three:

Example Four:

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