Derivative of e^x

In this tutorial we shall find the derivative of exponential function {e^x}. We prove the general rules for differentiation of exponential functions.
Let us suppose that the function of the form

y = f\left( x \right) = {e^x}

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

\begin{gathered} y + \Delta y = {e^{x + \Delta x}} \\ \Delta y = {e^{x + \Delta x}} - y \\ \end{gathered}

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

\Delta y = {e^{x + \Delta x}} - {e^x}

Dividing both sides by \Delta x, we get

\frac{{\Delta y}}{{\Delta x}} = \frac{{{e^{x + \Delta x}} - {e^x}}}{{\Delta x}}

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

\begin{gathered} \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{{e^{x + \Delta x}} - {e^x}}}{{\Delta x}} \\ \Rightarrow \frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{{e^x}\left( {{e^{\Delta x}} - 1} \right)}}{{\Delta x}} \\ \end{gathered}

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

\frac{{dy}}{{dx}} = {e^x}\ln e

But we have the relation \ln e = 1

\frac{d}{{dx}}{e^x} = {e^x}

Example: Find the derivative of

y = f\left( x \right) = {e^{\sin x}}

We have the given function as

y = {e^{\sin x}}

Differentiation with respect to variable x, we get

\frac{{dy}}{{dx}} = \frac{d}{{dx}}{e^{\sin x}}

Using the rule, \frac{d}{{dx}}{e^x} = {e^x}, we get

\begin{gathered} \frac{{dy}}{{dx}} = {e^{\sin x}}\frac{d}{{dx}}\left( {\sin x} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = {e^{\sin x}}\cos x \\ \end{gathered}