# Derivative of Exponential function formula

## Formula

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

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 $a^{\displaystyle x}$ and $\ln{a}$.

### Introduction

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

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

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

#### Other form

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

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

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

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

### Proof

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

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