# Derivative of Exponential Functions

In this tutorial we shall find the general rules of derivative of exponential functions, and we shall prove the general rules for the differentiation of exponential functions.

A function defined by $f$ where $f\left( x \right) = {a^x},\,\,\,a > 0,\,\,\,a \ne 1$ and $x$ is a real number is called an exponential function.

Let us suppose that the function is of the form $y = f\left( x \right) = {a^x}$, where $a > 0,\,\,a \ne 1$

First we take the increment or small change in the function:

Putting the value of function $y = {a^x}$ in the above equation, we get

Dividing both sides by $\Delta x$, we get

Taking the limit of both sides as $\Delta x \to 0$, we have

Using the following relation from the limit $\mathop {\lim }\limits_{x \to 0} \frac{{{a^x} - 1}}{x} = \ln a$, we have

Example: Find the derivative of

We have the given function as

Differentiating with respect to variable $x$, we get

Using the rule, $\frac{d}{{dx}}{a^x} = {a^x}\ln a$, we get