Power Rule

The rule

For any fixed exponent $n$ , the derivative of $x^n$ brings the exponent down as a multiplier and reduces it by one:

\dfrac{d}{dx}\left[x^n\right] = nx^{n-1}

This works for negative and fractional exponents too, not just positive whole numbers.

Every kind of exponent

The rule needs the function written as a single power of $x$ first. Roots become fractional powers and reciprocals become negative powers, then the same two steps apply: bring the power down, and subtract one from it.

The power rule applied to whole, zero, negative and fractional exponents

Function	Rewrite as	Derivative
$x^5$	$x^5$	$5x^4$
$x$	$x^1$	$1$
$7$ (a constant)	$7x^0$	$0$
$\dfrac{1}{x}$	$x^{-1}$	$-x^{-2} = -\dfrac{1}{x^2}$
$\dfrac{1}{x^3}$	$x^{-3}$	$-3x^{-4} = -\dfrac{3}{x^4}$
$\sqrt{x}$	$x^{1/2}$	$\dfrac{1}{2}x^{-1/2} = \dfrac{1}{2\sqrt{x}}$
$\sqrt[3]{x^2}$	$x^{2/3}$	$\dfrac{2}{3}x^{-1/3} = \dfrac{2}{3\sqrt[3]{x}}$

Worked example

Find the derivative of $f(x) = \sqrt{x}$ .

Solution: Rewrite the square root as a fractional exponent, $f(x) = x^{1/2}$ , then apply the power rule with $n = \frac{1}{2}$ :

f'(x) = \dfrac{1}{2}x^{\frac{1}{2} - 1} = \dfrac{1}{2}x^{-\frac{1}{2}}

Worked example: negative and fractional powers together

Differentiate $f(x) = \dfrac{4}{x^2} - 6\sqrt[3]{x} + 5$ .

Solution: Rewrite every term as a power of $x$ : $f(x) = 4x^{-2} - 6x^{1/3} + 5$ . Differentiate term by term. A constant multiplier stays in front, and the constant term 5 differentiates to 0:

f'(x) = 4(-2)x^{-3} - 6\cdot\dfrac{1}{3}x^{-2/3} + 0 = -\dfrac{8}{x^3} - \dfrac{2}{\sqrt[3]{x^2}}

What the rule looks like on a graph

y = x³ and its derivative y′ = 3x². The derivative's height at any x is the cubic's slope there: 3 at x = 1, 12 at x = 2, and 0 at x = 0.

The derivative of $x^3$ is $3x^2$ , which is never negative, so $x^3$ rises everywhere. Its slope drops to 0 at $x = 0$ , where the cubic flattens out for an instant, then grows again on both sides. Reading a derivative this way, as the slope of the original graph, is the idea behind increasing and decreasing functions.

Common mistakes

Moving the wrong way on a negative power. The derivative of $x^{-2}$ is $-2x^{-3}$ , not $-2x^{-1}$ : one less than $-2$ is $-3$ .
Differentiating before rewriting. $\dfrac{1}{x^3}$ does not become $\dfrac{1}{3x^2}$ . Rewrite it as $x^{-3}$ first, which gives $-\dfrac{3}{x^4}$ .
Using it on an exponential. The power rule needs a variable base and a fixed exponent. In $2^x$ the exponent is the variable, and its derivative is $2^x\ln 2$ , not $x\cdot 2^{x-1}$ .
Mixing up constant terms and constant multipliers. $\dfrac{d}{dx}(5x^3) = 15x^2$ , because the 5 stays as a multiplier, but $\dfrac{d}{dx}(5) = 0$ .