# Power Rule for Derivatives: Formula, Examples and Proof

> The power rule: the derivative of x^n is n·x^(n−1). Formula, examples with negative and fractional powers, a short proof and common mistakes.

Source: https://practicecalculus.com/rules/power-rule
Updated: 2026-11-03

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

## At a glance

- **Applies to:** Any power of $x$ with a constant exponent $n$ (positive, negative, fraction or decimal)
- **In words:** Bring the exponent down in front, then subtract one from the exponent
- **Does not apply to:** Exponentials like $2^x$ or $e^x$, where the variable is in the exponent
- **Integral version:** $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ for $n \ne -1$
- **Related rules:** Constant multiple rule, sum rule, chain rule

## Key takeaways

- The derivative of $x^n$ is $nx^{n-1}$ for any constant exponent $n$.
- Rewrite roots and fractions as powers first: $\sqrt{x} = x^{1/2}$ and $\frac{1}{x^3} = x^{-3}$.
- The power rule does not apply to exponential functions such as $2^x$.

The **power rule** is the first derivative shortcut most students learn, and the one they use most. It differentiates any power of $x$ in one step, with no limits required.

## The power rule formula

For any constant exponent $n$:

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

Two moves: **bring the exponent down** as a coefficient, then **subtract one** from the exponent.

```graph
title: 'Graph of y = x³ and its derivative y = 3x²'
caption: 'Where $x^3$ is steep, its derivative $3x^2$ is large. At $x = 0$ the curve is momentarily flat, and $3x^2 = 0$.'
x: [-2.2, 2.2]
y: [-6, 8]
functions:
  - expr: x^3
    label: y = x³
  - expr: 3x^2
    label: y = 3x² (derivative)
    dashed: true
```

## Examples

| Function | Rewrite | Derivative |
| --- | --- | --- |
| $x^5$ | $x^5$ | $5x^4$ |
| $x$ | $x^1$ | $1$ |
| $7$ | $7x^0$ | $0$ |
| $\frac{1}{x^2}$ | $x^{-2}$ | $-2x^{-3} = -\frac{2}{x^3}$ |
| $\sqrt{x}$ | $x^{1/2}$ | $\frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$ |
| $\sqrt[3]{x^2}$ | $x^{2/3}$ | $\frac{2}{3}x^{-1/3}$ |

**Example: a polynomial.** Combine the power rule with the [sum rule](/rules/sum-and-difference-rule) and [constant multiple rule](/rules/constant-multiple-rule):

$$\frac{d}{dx}\left(4x^3 - 5x^2 + 2x - 9\right) = 12x^2 - 10x + 2$$

**Example: a negative power.** To differentiate $f(x) = \frac{3}{x^4}$, rewrite it as $3x^{-4}$:

$$f'(x) = 3 \cdot (-4)x^{-5} = -\frac{12}{x^5}$$

Notice that the exponent goes from $-4$ to $-5$, not to $-3$. Subtracting one from a negative number makes it more negative.

## When to use the power rule

Use it whenever you see a variable raised to a **constant** power, after any rewriting. In practice that means:

- polynomials, one term at a time;
- roots, after converting them to fractional exponents;
- terms with $x$ in the denominator, after converting them to negative exponents;
- the outer layer of a function like $(x^2+1)^5$, together with the chain rule.

If the exponent contains $x$, as in $x^x$ or $3^x$, the power rule is the wrong tool. Those need exponential rules or logarithmic differentiation.

## The power rule in reverse: integration

Integration undoes differentiation, so the power rule has a mirror image. To integrate $x^n$, **add one** to the exponent, then **divide** by the new exponent:

$$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \qquad n \ne -1$$

For example, $\int x^3\,dx = \frac{x^4}{4} + C$. The single exception is $n = -1$, where $\int x^{-1}\,dx = \ln|x| + C$. See the [reverse power rule](/rules/reverse-power-rule).

## Why the power rule works

For a whole-number exponent, expand $(x+h)^n$ with the binomial theorem inside the [limit definition of the derivative](/blog/limit-definition-of-the-derivative):

$$\frac{(x+h)^n - x^n}{h} = \frac{nx^{n-1}h + \binom{n}{2}x^{n-2}h^2 + \cdots + h^n}{h} = nx^{n-1} + h(\ldots)$$

As $h \to 0$, every term with an $h$ in it vanishes, leaving $nx^{n-1}$. The rule extends to all real exponents using logarithmic differentiation.

## Common mistakes

1. **Subtracting one before bringing the exponent down.** $\frac{d}{dx}x^4$ is $4x^3$, not $3x^3$.
2. **Using it on exponentials.** $\frac{d}{dx}2^x \ne x \cdot 2^{x-1}$. The variable is in the exponent, so you need the [exponential rule](/blog/derivatives-of-exponential-and-logarithmic-functions).
3. **Not rewriting first.** You cannot apply the power rule to $\frac{1}{x^2}$ until you write it as $x^{-2}$.
4. **Forgetting the chain rule.** For $(3x+1)^4$ the power rule gives only part of the answer. You also multiply by the inside derivative, $3$. See the [chain rule](/rules/chain-rule).

For a longer walkthrough with more practice problems, read the full guide to [the power rule](/blog/power-rule).

## Frequently asked questions

### What is the power rule in calculus?

The power rule is the derivative shortcut $\frac{d}{dx}x^n = nx^{n-1}$. You multiply by the exponent and then lower the exponent by one.

### Does the power rule work for negative and fractional exponents?

Yes. It works for every real constant exponent. For example $\frac{d}{dx}x^{-2} = -2x^{-3}$ and $\frac{d}{dx}x^{1/2} = \frac{1}{2}x^{-1/2}$.

### What is the derivative of a constant using the power rule?

A constant $c$ can be written as $cx^0$, and the power rule gives $0 \cdot cx^{-1} = 0$. The derivative of any constant is zero.

### Why does the power rule not work for 2^x?

In $2^x$ the variable is in the exponent, so it is an exponential function, not a power. Its derivative is $2^x \ln 2$.