# Derivative of Exponential Function

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

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

Let us suppose that the function 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 limit of both sides as $\Delta x \to 0$, we have

Using the following relation from 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

Differentiation with respect to variable $x$, we get

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