# Derivative of Exponential function

## Formula

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

The derivative of an exponential function is equal to the product of the exponential function and natural logarithm of the base of exponential function. It is called the derivative rule of exponential function.

### Introduction

Suppose $a$ and $x$ represent a constant and a variable respectively then the exponential function is written as $a^{\displaystyle x}$ in mathematics. The derivative of $a$ raised to the power of $x$ with respect to $x$ is written in the following form in calculus.

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

In differential calculus, the differentiation or derivative 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.

The derivative of $a$ raised to the $x$-th power with respect to $x$ is equal to the product of $a$ to the $x$-th power and the natural logarithm of $a$.

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

It is called the differentiation rule of exponential function and it is used to find the derivative of any exponential function.

#### Example

Evaluate $\dfrac{d}{dx}{\, \big(5^{\displaystyle x}\big)}$

In this example, the constant $a \,=\, 5$. Now, substitute it in the differentiation law of exponential function to find its derivative.

$=\,\,\,$ $5^{\displaystyle x}\log_{e}{(5)}$

Thus, it can be used as a formula to find the differentiation of any function in exponential form.

#### Other forms

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

$(1).\,\,\,$ $\dfrac{d}{dy}{\, (c^{\displaystyle y})} \,=\, c^{\displaystyle y}\log_{e}{c}$

$(2).\,\,\,$ $\dfrac{d}{dv}{\, (k^{\displaystyle v})} \,=\, k^{\displaystyle v}\log_{e}{k}$

$(3).\,\,\,$ $\dfrac{d}{dz}{\, (u^{\displaystyle z})} \,=\, u^{\displaystyle z}\log_{e}{u}$

### Proofs

The derivative rule of exponential function can be proved in the following two methods.

#### Fundamental

Learn how to prove the derivative of $a$ raised to the $x$-th power with respect to $x$ fundamentally from the first principle of derivatives.

#### Differentiation

Learn how to derive the derivative of the $x$-th power of $a$ with respect to $x$ by using both logarithm and differentiation rules.

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