Power Rule for Derivatives: Examples, Proof & Live Check

Q: What is the power rule for derivatives?

\frac{d}{dx}x^n = n\,x^{n-1}: bring the exponent down in front, then subtract one from the exponent. It works for every real n: whole numbers, fractions and negatives.

Q: How do I use it on roots and fractions?

Rewrite them as powers first. \sqrt{x} = x^{1/2} gives \frac12 x^{-1/2} = \frac{1}{2\sqrt{x}}. \frac{1}{x^3} = x^{-3} gives -3x^{-4}.

Q: What is the power rule for integrals?

The reverse: \int x^n\,dx = \frac{x^{n+1}}{n+1} + C for n \ne -1. Add one to the exponent, then divide by the new exponent. For n = -1 the answer is \ln|x| + C.

Q: Why does the power rule work?

For whole numbers, expand (x+h)^n = x^n + n x^{n-1} h + (\text{terms with } h^2). Subtract x^n, divide by h, let h \to 0: only n x^{n-1} survives.

\frac{d}{dx}\,x^n = n\,x^{n-1}

Check it yourself

The graph measures the slope of $x^n$ numerically, as rise over a tiny run, and compares it with what the rule predicts. Change $n$ , drag the point, and see if you can break it:

Exponent n = 3

Measured slope at x = 1.20: 4.32

Power rule $n x^{n-1} = 3 \cdot 1.20^{2}$ = 4.32

Try fractional and negative n: the rule keeps matching the measured slope.

Worked examples

Function	Rewrite as a power	Derivative
$x^5$	$x^5$	$5x^4$
$7x^3$	$7x^3$	$21x^2$
$\sqrt{x}$	$x^{1/2}$	$\tfrac12 x^{-1/2} = \tfrac{1}{2\sqrt x}$
$\frac{1}{x^2}$	$x^{-2}$	$-2x^{-3} = -\tfrac{2}{x^3}$
$\sqrt[3]{x^2}$	$x^{2/3}$	$\tfrac23 x^{-1/3}$
$4x^{-1/2}$	$4x^{-1/2}$	$-2x^{-3/2}$

Why it works

From the definition of the derivative with $n = 3$ :

\frac{(x+h)^3 - x^3}{h} = \frac{3x^2h + 3xh^2 + h^3}{h} = 3x^2 + 3xh + h^2 \;\to\; 3x^2

For any whole $n$ , the binomial theorem gives $(x+h)^n = x^n + nx^{n-1}h + \dots$ , and every term after the second has $h^2$ or more, so it vanishes. For fractional and negative powers the same result follows from implicit differentiation or from $x^n = e^{n\ln x}$ .

The power rule for integrals

Run it backwards: $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ , except $n = -1$ , where it would divide by zero and the answer is $\ln|x| + C$ instead. More on integration rules.

Questions students ask

What is the power rule for derivatives?

$\frac{d}{dx}x^n = n\,x^{n-1}$ : bring the exponent down in front, then subtract one from the exponent. It works for every real $n$ : whole numbers, fractions and negatives.

How do I use it on roots and fractions?

Rewrite them as powers first. $\sqrt{x} = x^{1/2}$ gives $\frac12 x^{-1/2} = \frac{1}{2\sqrt{x}}$ . $\frac{1}{x^3} = x^{-3}$ gives $-3x^{-4}$ .

What is the power rule for integrals?

The reverse: $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ for $n \ne -1$ . Add one to the exponent, then divide by the new exponent. For $n = -1$ the answer is $\ln|x| + C$ .

Why does the power rule work?

For whole numbers, expand $(x+h)^n = x^n + n x^{n-1} h + (\text{terms with } h^2)$ . Subtract $x^n$ , divide by $h$ , let $h \to 0$ : only $n x^{n-1}$ survives.

The Power Rule

Check it yourself

Worked examples

Why it works

The power rule for integrals

Still stuck? Ask the tutor

Questions students ask