# Power Rule of Derivatives

## Formula

$\dfrac{d}{dx}{\, (x^{\displaystyle n})}$ $\,=\,$ $n.x^{\, \displaystyle n\small -1}$

The derivative of an exponential term, which contains a variable as base and a constant as power, is called the constant power derivative rule.

### Introduction

$x$ and $n$ are literals and they represent a variable and a constant. They form an exponential term $x^n$. The derivative of $x$ is raised to the power $n$ is written in mathematical form as follows.

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

The differentiation of $n$-th power of $x$ with respect to $x$ is equal to the product of $n$ and $x$ raised to the power of $n$ minus one.

$\dfrac{d}{dx}{\, (x^{\displaystyle n})}$ $\,=\,$ $n.x^{\, \displaystyle n\small -1}$

#### Alternative form

The derivative of a constant power rule can be written in terms of any variable and constant.

$(1) \,\,\,\,\,\,$ $\dfrac{d}{dh}{\, (h^{\displaystyle c})}$ $\,=\,$ $c.h^{\, \displaystyle c\small -1}$

$(2) \,\,\,\,\,\,$ $\dfrac{d}{dl}{\, (l^{\displaystyle m})}$ $\,=\,$ $m.l^{\, \displaystyle m\small -1}$

$(3) \,\,\,\,\,\,$ $\dfrac{d}{dy}{\, (y^{\displaystyle p})}$ $\,=\,$ $p.y^{\, \displaystyle p\small -1}$

### Proof

Learn how to prove the derivative of a constant power rule in differential calculus by first principle.

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Jun 26, 2023

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