# The Power Rule for Derivatives: Rules, Examples and Tricks

> Master the power rule: how it works, negative and fractional exponents, rewriting roots and fractions, plus the sum and constant multiple rules.

Source: https://practicecalculus.com/blog/power-rule
Updated: 2026-08-17

## Key takeaways

- The power rule says $\frac{d}{dx} x^n = n x^{n-1}$: bring the exponent down, then subtract one from it.
- It works for every real exponent, including negative numbers and fractions.
- Rewrite roots and fractions as powers first: $\sqrt{x} = x^{1/2}$ and $\frac{1}{x^3} = x^{-3}$.
- Constants multiply through, sums differentiate term by term, and the derivative of a constant is $0$.

Computing derivatives with the limit definition means a page of algebra for every function. The **power rule** replaces all of that with a one-line shortcut, and combined with two simple companion rules, it lets you differentiate any polynomial, root or power of $x$ in seconds. It is the first rule you learn and the one you will use most.

## What is the power rule?

The power rule tells you the derivative of $x$ raised to a constant power:

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

In words: **bring the exponent down in front, then subtract one from the exponent.**

$$\frac{d}{dx} x^5 = 5x^4 \qquad \frac{d}{dx} x^2 = 2x \qquad \frac{d}{dx} x = 1$$

The rule works for every real number $n$: positive whole numbers, negative numbers and fractions. The only requirement is that the variable is in the base and the exponent is a constant.

## Why the power rule works

For a positive whole number $n$, you can see the pattern from the [limit definition of the derivative](/blog/limit-definition-of-the-derivative). Expanding $(x + h)^n$ gives

$$(x + h)^n = x^n + n x^{n-1} h + (\text{terms with } h^2 \text{ or higher})$$

So

$$\frac{(x+h)^n - x^n}{h} = n x^{n-1} + (\text{terms with } h \text{ or higher})$$

and as $h \to 0$ everything except $n x^{n-1}$ disappears. That is the power rule. For $n = 2$ this is exactly the familiar calculation $(x+h)^2 = x^2 + 2xh + h^2$, which gives $2x$. The proof for negative and fractional exponents uses other tools, but the rule stays the same.

## The constant, constant multiple and sum rules

The power rule rarely appears alone. Three companion rules let you combine it with coefficients and multiple terms.

| Rule | Formula | Example |
| --- | --- | --- |
| Constant rule | $\frac{d}{dx} c = 0$ | $\frac{d}{dx} 7 = 0$ |
| Constant multiple rule | $\frac{d}{dx} [c f(x)] = c f'(x)$ | $\frac{d}{dx} 4x^3 = 12x^2$ |
| Sum and difference rule | $\frac{d}{dx} [f \pm g] = f' \pm g'$ | $\frac{d}{dx} (x^2 + x) = 2x + 1$ |

The constant rule says a horizontal line has slope $0$. The constant multiple rule says stretching a graph vertically by $c$ multiplies every slope by $c$. The sum rule says you can differentiate a polynomial one term at a time.

## Worked example 1: a polynomial

Differentiate $f(x) = 3x^4 - 5x^2 + 7x - 2$.

**Step 1. Split into terms.** By the sum rule, differentiate each term separately.

**Step 2. Apply the power rule with constant multiples.**

- $3x^4 \to 3 \cdot 4x^3 = 12x^3$
- $-5x^2 \to -5 \cdot 2x = -10x$
- $7x \to 7 \cdot 1 = 7$
- $-2 \to 0$

**Step 3. Combine.**

$$f'(x) = 12x^3 - 10x + 7$$

## Negative exponents: rewriting fractions

A power of $x$ in a denominator is a negative exponent in disguise:

$$\frac{1}{x^n} = x^{-n}$$

Rewrite first, then apply the power rule. Pay close attention when you subtract one: the exponent becomes **more** negative.

### Worked example 2: a fraction with a coefficient

Differentiate $y = \frac{4}{x^3}$.

**Step 1. Rewrite as a power.** $y = 4x^{-3}$.

**Step 2. Apply the power rule.** Bring down $-3$ and subtract one: $-3 - 1 = -4$.

$$\frac{dy}{dx} = 4 \cdot (-3) x^{-4} = -12x^{-4}$$

**Step 3. Rewrite without negative exponents** (optional, but usually expected).

$$\frac{dy}{dx} = -\frac{12}{x^4}$$

A special case is worth memorizing: $\frac{d}{dx} \frac{1}{x} = \frac{d}{dx} x^{-1} = -x^{-2} = -\frac{1}{x^2}$, the same result the limit definition gives.

> **Tip:** watch where the coefficient lives. $\frac{1}{3x^2}$ is $\frac{1}{3}x^{-2}$, but $\frac{3}{x^2}$ is $3x^{-2}$. Only the power of $x$ moves up to the numerator as a negative exponent.

## Fractional exponents: rewriting roots

Roots are fractional powers:

$$\sqrt{x} = x^{1/2} \qquad \sqrt[3]{x} = x^{1/3} \qquad \sqrt[n]{x^m} = x^{m/n}$$

### Worked example 3: a square root

Differentiate $y = \sqrt{x}$.

**Step 1. Rewrite.** $y = x^{1/2}$.

**Step 2. Apply the power rule.** $\frac{1}{2} - 1 = -\frac{1}{2}$, so

$$\frac{dy}{dx} = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}$$

### Worked example 4: a root of a power

Differentiate $y = \sqrt[3]{x^2}$.

**Step 1. Rewrite.** $y = x^{2/3}$.

**Step 2. Apply the power rule.** $\frac{2}{3} - 1 = -\frac{1}{3}$, so

$$\frac{dy}{dx} = \frac{2}{3} x^{-1/3} = \frac{2}{3\sqrt[3]{x}}$$

Notice the derivative is undefined at $x = 0$. The graph of $x^{2/3}$ has a sharp cusp there.

## Worked example 5: simplify before you differentiate

Differentiate $y = \frac{x^2 + 1}{\sqrt{x}}$.

You could use the [quotient rule](/blog/product-and-quotient-rules), but splitting the fraction is faster.

**Step 1. Split into separate terms.**

$$y = \frac{x^2}{x^{1/2}} + \frac{1}{x^{1/2}}$$

**Step 2. Subtract exponents.** $x^{2 - 1/2} = x^{3/2}$, so

$$y = x^{3/2} + x^{-1/2}$$

**Step 3. Apply the power rule to each term.**

$$\frac{dy}{dx} = \frac{3}{2}x^{1/2} - \frac{1}{2}x^{-3/2}$$

**Step 4. Rewrite with roots** (optional).

$$\frac{dy}{dx} = \frac{3\sqrt{x}}{2} - \frac{1}{2x\sqrt{x}}$$

The same trick works for products: expand $(2x + 1)^2 = 4x^2 + 4x + 1$, then differentiate to get $8x + 4$.

## Putting it all together

Differentiate $y = 6\sqrt[3]{x^5} - \frac{2}{x} + 9$.

**Step 1. Rewrite every term as a power of $x$.**

$$y = 6x^{5/3} - 2x^{-1} + 9$$

**Step 2. Apply the power rule term by term.** For the first term, $\frac{5}{3} - 1 = \frac{2}{3}$. For the second, $-1 - 1 = -2$. The constant $9$ becomes $0$.

$$\frac{dy}{dx} = 6 \cdot \frac{5}{3} x^{2/3} - 2 \cdot (-1) x^{-2} = 10x^{2/3} + 2x^{-2}$$

**Step 3. Rewrite in root and fraction form.**

$$\frac{dy}{dx} = 10\sqrt[3]{x^2} + \frac{2}{x^2}$$

Notice how the two minus signs in the middle term combined into a plus. Sign slips like that are the main thing to watch once you are comfortable with the rule itself.

## Worked example 6: a tangent line

Find the equation of the tangent line to $y = x^3$ at $x = 2$.

**Step 1. Find the point.** $y(2) = 2^3 = 8$, so the point is $(2, 8)$.

**Step 2. Find the slope.** $\frac{dy}{dx} = 3x^2$, so the slope at $x = 2$ is $3(2)^2 = 12$.

**Step 3. Use point-slope form.**

$$y - 8 = 12(x - 2) \quad \Longrightarrow \quad y = 12x - 16$$

```graph
title: 'Graph of y = x³ with its tangent line y = 12x − 16 at the point (2, 8)'
caption: 'By the power rule the slope of $y = x^3$ at $x = 2$ is $3(2)^2 = 12$, giving the tangent line $y = 12x - 16$.'
x: [-3, 3.5]
y: [-12, 20]
functions:
  - expr: x^3
    label: y = x³
  - expr: 12x - 16
    label: tangent at x = 2
    dashed: true
points:
  - [2, 8, '(2, 8)']
```

## Higher derivatives with the power rule

Because the derivative of a polynomial is another polynomial, you can apply the power rule again and again. The **second derivative** $f''(x)$ is the derivative of $f'(x)$, and it measures how the slope itself is changing (concavity).

### Worked example 7: first, second and third derivatives

Find $f'$, $f''$ and $f'''$ for $f(x) = 2x^5 - x^3 + 4x$.

**Step 1. First derivative.** Apply the power rule term by term:

$$f'(x) = 10x^4 - 3x^2 + 4$$

**Step 2. Second derivative.** Differentiate $f'$. The constant $4$ disappears:

$$f''(x) = 40x^3 - 6x$$

**Step 3. Third derivative.**

$$f'''(x) = 120x^2 - 6$$

Each round lowers every exponent by one, so a polynomial of degree $n$ becomes $0$ after $n + 1$ derivatives.

## The power rule as a rate of change

Derivatives are rates, so the power rule turns position formulas into velocity formulas instantly. Suppose a ball's height in meters after $t$ seconds is

$$s(t) = -5t^2 + 20t + 2$$

The variable is $t$ instead of $x$, but the rule is the same. The velocity is

$$v(t) = s'(t) = -10t + 20$$

and the acceleration is $a(t) = v'(t) = -10$ meters per second squared. Setting $v(t) = 0$ gives $t = 2$, the moment the ball stops rising, at a height of $s(2) = -20 + 40 + 2 = 22$ meters. Every one of those answers came from the power rule and nothing else.

## Common mistakes with the power rule

1. **Subtracting one wrong with negative exponents.** $\frac{d}{dx} x^{-3} = -3x^{-4}$. Going from $-3$ to $-2$ is the most common error.
2. **Differentiating a fraction piece by piece.** $\frac{d}{dx} \frac{1}{x^3}$ is not $\frac{1}{3x^2}$. Rewrite as $x^{-3}$ first.
3. **Forgetting that constants become $0$.** The derivative of $-2$, $\pi^2$ or $e^3$ is $0$, not the constant itself.
4. **Applying the power rule to exponentials.** $\frac{d}{dx} 2^x$ is not $x \cdot 2^{x-1}$. The variable is in the exponent, so the power rule does not apply.
5. **Using it on a power of something other than $x$.** $\frac{d}{dx} (3x+1)^5$ needs the [chain rule](/blog/chain-rule), which adds a factor of $3$.
6. **Moving a coefficient with the power.** $\frac{5}{x^2} = 5x^{-2}$, not $5^{-1}x^{-2}$.

## Quick reference

| Function | Rewrite as | Derivative |
| --- | --- | --- |
| $x^n$ | $x^n$ | $n x^{n-1}$ |
| $c$ (constant) | | $0$ |
| $x$ | $x^1$ | $1$ |
| $\frac{1}{x}$ | $x^{-1}$ | $-\frac{1}{x^2}$ |
| $\frac{1}{x^2}$ | $x^{-2}$ | $-\frac{2}{x^3}$ |
| $\sqrt{x}$ | $x^{1/2}$ | $\frac{1}{2\sqrt{x}}$ |
| $\sqrt[3]{x}$ | $x^{1/3}$ | $\frac{1}{3\sqrt[3]{x^2}}$ |
| $x\sqrt{x}$ | $x^{3/2}$ | $\frac{3}{2}\sqrt{x}$ |

## Practice until it is automatic

The power rule itself is easy. The skill that takes practice is the rewriting: turning roots and fractions into powers, and handling negative and fractional exponents without slipping. The interactive lesson below gives you a steady stream of problems, checks every answer instantly, and shows you exactly which exponent went wrong. A short session today will make every later derivative faster.

## Frequently asked questions

### What is the power rule in calculus?

The power rule says the derivative of $x^n$ is $n x^{n-1}$. Bring the exponent down in front as a coefficient, then lower the exponent by one. For example, the derivative of $x^5$ is $5x^4$.

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

Yes. For example, $\frac{d}{dx} x^{-3} = -3x^{-4}$. Be careful when subtracting one from a negative exponent: $-3 - 1 = -4$, not $-2$.

### How do you use the power rule on a square root?

Rewrite the root as a fractional power, then apply the rule. $\sqrt{x} = x^{1/2}$, so its derivative is $\frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$.

### What is the derivative of a constant?

The derivative of any constant is $0$, because a constant function is a horizontal line with slope $0$. This includes constants like $7$, $\pi^2$ and $e^3$.

### Can you use the power rule on e^x or 2^x?

No. The power rule needs the variable in the base and a constant in the exponent, like $x^2$. In $e^x$ and $2^x$ the variable is in the exponent, so they use exponential rules: $\frac{d}{dx} e^x = e^x$ and $\frac{d}{dx} 2^x = 2^x \ln 2$.