Math Doubts

Power Rule of Derivatives


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

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


$x$ 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.

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

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

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


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

Math Doubts
Math Doubts is a free math tutor for helping students to learn mathematics online from basics to advanced scientific level for teachers to improve their teaching skill and for researchers to share their research projects. Know more
Follow us on Social Media
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more