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:
\[\begin{gathered} y + \Delta y = {a^{x + \Delta x}} \\ \Delta y = {a^{x + \Delta x}} – y \\ \end{gathered} \]

Putting the value of function $$y = {a^x}$$ in the above equation, we get
\[\Delta y = {a^{x + \Delta x}} – {a^x}\]

Dividing both sides by $$\Delta x$$, we get
\[\frac{{\Delta y}}{{\Delta x}} = \frac{{{a^{x + \Delta x}} – {a^x}}}{{\Delta x}}\]

Taking the limit of both sides as $$\Delta x \to 0$$, we have
\[\begin{gathered} \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{{a^{x + \Delta x}} – {a^x}}}{{\Delta x}} \\ \Rightarrow \frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{{a^x}\left( {{a^{\Delta x}} – 1} \right)}}{{\Delta x}} \\ \end{gathered} \]

Using the following relation from the limit $$\mathop {\lim }\limits_{x \to 0} \frac{{{a^x} – 1}}{x} = \ln a$$, we have
\[\begin{gathered} \frac{{dy}}{{dx}} = {a^x}\ln a \\ \Rightarrow \frac{d}{{dx}}{a^x} = {a^x}\ln a \\ \end{gathered} \]

Example: Find the derivative of \[y = f\left( x \right) = {4^{2{x^3}}}\]

We have the given function as
\[y = {4^{2{x^3}}}\]

Differentiating with respect to variable $$x$$, we get
\[\frac{{dy}}{{dx}} = \frac{d}{{dx}}{4^{2{x^3}}}\]

Using the rule, $$\frac{d}{{dx}}{a^x} = {a^x}\ln a$$, we get
\[\begin{gathered} \frac{{dy}}{{dx}} = {4^{2{x^3}}}\ln 4\frac{d}{{dx}}\left( {2{x^3}} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = {4^{2{x^3}}}\ln 4\left( {6{x^2}} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = 6{x^2}{4^{2{x^3}}}\ln 4 \\ \end{gathered} \]