# Power Functions f(x) = xⁿ: Graphs, Derivative, Integral and Rules

> Power functions f(x) = xⁿ for whole, negative and fractional n: graphs, domains, the derivative nxⁿ⁻¹, the integral and the n = -1 exception.

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

$$ f(x) = x^n $$

## At a glance

- **Domain:** All reals for whole $n \ge 0$; $x \ne 0$ for negative $n$
- **Derivative:** $nx^{n-1}$
- **Integral:** $\frac{x^{n+1}}{n+1} + C$
- **Exception:** $\int x^{-1}\,dx = \ln|x| + C$
- **Even or odd:** Even for even whole $n$, odd for odd whole $n$
- **Passes through:** $(1, 1)$ for every $n$

## Key takeaways

- A power function is $f(x) = x^n$ with a constant exponent $n$, which may be a whole number, negative or a fraction.
- The derivative is $nx^{n-1}$ for every real constant $n$.
- The integral is $\frac{x^{n+1}}{n+1} + C$, except when $n = -1$, where it is $\ln|x| + C$.
- Rewrite roots and reciprocals as powers before differentiating or integrating.

A **power function** raises the input to a fixed exponent: $f(x) = x^n$. The family includes parabolas, cubics, reciprocals and roots, and one rule, the [power rule](/rules/power-rule), differentiates every member.

## Formula

$$f(x) = x^n$$

The exponent $n$ is a constant. More generally $f(x) = kx^n$ with a constant coefficient $k$. Common members:

| $n$ | Function | Name |
| --- | --- | --- |
| $0$ | $1$ | [constant](/functions/constant-function) |
| $1$ | $x$ | identity line |
| $2$ | $x^2$ | parabola |
| $3$ | $x^3$ | cubic |
| $-1$ | $\frac{1}{x}$ | [reciprocal](/functions/reciprocal-function) |
| $\frac{1}{2}$ | $\sqrt{x}$ | [square root](/functions/square-root-function) |

## Graphs of power functions

```graph
title: 'Graphs of the power functions y = x², y = x³ and y = √x'
caption: 'Every power function passes through $(1, 1)$. Even powers are symmetric about the $y$-axis, odd powers are symmetric about the origin.'
x: [-2.5, 2.5]
y: [-3, 4]
functions:
  - expr: x^2
    label: y = x²
  - expr: x^3
    label: y = x³
  - expr: sqrt(x)
    label: y = √x
points:
  - [1, 1, '(1, 1)']
```

How the exponent shapes the graph:

- **Even whole $n$** ($x^2, x^4, \ldots$): U-shaped, even function, range $[0, \infty)$.
- **Odd whole $n$** ($x^3, x^5, \ldots$): S-shaped, odd function, range all reals.
- **Negative $n$** ($x^{-1}, x^{-2}, \ldots$): undefined at $0$, with a vertical asymptote at $x = 0$ and a horizontal asymptote $y = 0$.
- **Fractional $n = \frac{p}{q}$**: an even root ($q$ even) restricts the domain to $x \ge 0$; an odd root allows all reals.
- On $(0, 1)$ higher powers are smaller; on $(1, \infty)$ higher powers are larger.

## Derivative of a power function

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

Bring the exponent down, then subtract one. This holds for every real constant $n$ wherever $x^n$ is differentiable. Note that for $0 < n < 1$, such as $\sqrt{x}$, the derivative blows up at $x = 0$, where the graph has a vertical tangent.

## Integral of a power function

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

Add one to the exponent, then divide by the new exponent. The case $n = -1$ would divide by zero; instead

$$\int \frac{1}{x}\,dx = \ln|x| + C$$

See the [reverse power rule](/rules/reverse-power-rule).

## Worked examples

**Example 1.** $\frac{d}{dx}x^7 = 7x^6$.

**Example 2: negative power.** $\frac{d}{dx}\frac{4}{x^3} = \frac{d}{dx}4x^{-3} = -12x^{-4} = -\frac{12}{x^4}$.

**Example 3: fractional power.** $\frac{d}{dx}\sqrt[3]{x} = \frac{d}{dx}x^{1/3} = \frac{1}{3}x^{-2/3}$.

**Example 4: integral.** $\int x^{3/2}\,dx = \frac{x^{5/2}}{5/2} + C = \frac{2}{5}x^{5/2} + C$.

**Example 5: definite integral.** $\int_1^4 \frac{1}{\sqrt{x}}\,dx = \left[2x^{1/2}\right]_1^4 = 4 - 2 = 2$.

## Common mistakes

1. **Using the power rule on $2^x$.** That is an [exponential function](/functions/exponential-functions); its derivative is $2^x\ln 2$.
2. **Subtracting one from a negative exponent in the wrong direction.** $x^{-3}$ differentiates to $-3x^{-4}$, not $-3x^{-2}$.
3. **Applying the integral formula at $n = -1$.** $\frac{x^0}{0}$ is meaningless; use $\ln|x|$.
4. **Not rewriting first.** Turn $\frac{1}{x^2}$ into $x^{-2}$ and $\sqrt{x}$ into $x^{1/2}$ before using either rule.

## How it connects

Sums of power functions with whole exponents are [polynomial functions](/functions/polynomial-functions), and power series extend that idea to infinitely many terms. For a deeper walkthrough with practice problems, read the guide to the [power rule](/blog/power-rule) and to [antiderivatives](/blog/antiderivatives).

## Frequently asked questions

### What is a power function?

A power function has the form $f(x) = x^n$ (or $kx^n$) where the exponent $n$ is a constant. Examples are $x^3$, $x^{-1} = \frac{1}{x}$ and $x^{1/2} = \sqrt{x}$.

### What is the derivative of x^n?

By the power rule, $\frac{d}{dx}x^n = nx^{n-1}$. This holds for any real constant $n$, including negative and fractional exponents.

### What is the integral of x^n?

For $n \ne -1$, $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$. When $n = -1$ the formula would divide by zero, and the integral is $\ln|x| + C$ instead.

### What is the difference between a power function and an exponential function?

In a power function such as $x^2$ the variable is the base and the exponent is fixed. In an exponential function such as $2^x$ the base is fixed and the variable is the exponent, and the derivative rules are completely different.