# Derivative of Exponential function formula

## Formula

$\dfrac{d}{dx}{\, (a^x})} \,=\, a^x}\log_{e}{a$

The differentiation of exponential function with respect to a variable is equal to the product of exponential function and natural logarithm of base of exponential function. It is read as the derivative of $a$ raised to the power of $x$ with respect to $x$ is equal to the product of $a^x$ and $\ln{a}$.

### Introduction

Take, $a$ is a constant and $x$ is a variable, then the exponential function is written as $a^x$ in mathematical form. The derivative of the $a^x$ function with respect to $x$ is written mathematically as follows.

$\dfrac{d}{dx}{\, (a^x})$

In differential calculus, the differentiation of the $a^x$ function with respect to $x$ is also written as $\dfrac{d{\,(a^x})}}{dx$ and is also simply written as ${(a^x})}$ in mathematics.

#### Other form

The formula for the derivative of exponential function can be written in terms of any variable.

$(1) \,\,\,$ $\dfrac{d}{ds}{\, (c^s})} \,=\, c^s}\log_{e}{c$

$(2) \,\,\,$ $\dfrac{d}{dl}{\, (g^l})} \,=\, g^l}\log_{e}{l$

$(3) \,\,\,$ $\dfrac{d}{dy}{\, (m^y})} \,=\, m^y}\log_{e}{m$

### Proof

Learn how to derive the differentiation of $a^x$ formula with respect to $x$ in differential calculus from first principle.

Email subscription
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more