# Derivative of e^x

In this tutorial we shall find the derivative of exponential function ${e^x}$ and we shall prove the general rules for the differentiation of exponential functions.

Let us suppose that the function is 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 the 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 the limit $\mathop {\lim }\limits_{x \to 0} \frac{{{a^x} – 1}}{x} = \ln a$, we have
$\frac{{dy}}{{dx}} = {e^x}\ln e$

Now 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
$y = {e^{\sin x}}$

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