# Math Insight

### The derivative of a power function

#### Video Introduction

The derivative of a power function.

#### Summary

For a power function $$f(x)=x^p,$$ with exponent $p \ne 0$, its derivative is \begin{align} f'(x) = \diff{f}{x} = p x^{p-1}. \end{align} (For fractional $p$, we may need to restrict the domain to positive numbers, $x > 0$, so that the function is real valued.)

Using this formula, we calculate derivatives for small positive and negative powers as well as some fractional powers.

Derivatives for some positive integer powers
$p$$f(x)$$f'(x)$ or $\diff{f}{x}$
$1$$x$$1$
$2$$x^2$$2x$
$3$$x^3$$3x^2$
$4$$x^4$$4x^3$
Derivatives for some negative integer powers
$p$$f(x)$$f'(x)$ or $\diff{f}{x}$
$-1$$x^{-1}=\frac{1}{x}$$-x^{-2}=-\frac{1}{x^2}$
$-2$$x^{-2}=\frac{1}{x^2}$$-2x^{-3}=-\frac{2}{x^3}$
$-3$$x^{-3}=\frac{1}{x^3}$$-3x^{-4}=-\frac{3}{x^4}$
$-4$$x^{-4}=\frac{1}{x^4}$$-4x^{-5}=-\frac{4}{x^5}$
Derivatives for some fractional powers
$p$$f(x)$$f'(x)$ or $\diff{f}{x}$
$\frac{1}{2}$$x^{1/2}=\sqrt{x}$$\frac{1}{2}x^{-1/2}=\frac{1}{2\sqrt{x}}$
$\frac{1}{3}$$x^{1/3}=\sqrt[3]{x}$$\frac{1}{3}x^{-2/3}=\frac{1}{3\sqrt[3]{x^2}}$
$-\frac{1}{2}$$x^{-1/2}=\frac{1}{\sqrt{x}}$$-\frac{1}{2}x^{-3/2}=-\frac{1}{2\sqrt{x^3}}$