# Power rule

The derivative of x^n is n x^(n-1). Constants disappear, and you differentiate sums term by term.

Maths (university) · Part 03: Differentiation · Card 4 of 10 · Last updated 2026-10-19

## Analogy

Bring the power down to the front, then lower the power by one.

## Example

**f(x) = 4x^3 - 5x + 7**

4x^3 becomes 12x^2.
-5x becomes -5.
The constant 7 becomes 0.
f'(x) = 12x^2 - 5.

## In practice

The fastest way to differentiate any polynomial.

## Practice

### Flashcards

- **What is d/dx of 4x^3 - 5x + 7?** 12x^2 - 5.

## Questions people ask

### Does the power rule work for fractional powers?

Yes. The rule holds for any real power n. The square root of x is x to the power 1/2, so its derivative is (1/2) x to the power -1/2, which is 1 over 2 times the square root of x. Rewrite roots and reciprocals as powers first.

### Why does the derivative of a constant equal zero?

A constant does not change, so its rate of change is zero. Its graph is a flat horizontal line with slope zero. That is why the +7 in 4x cubed - 5x + 7 vanishes when you differentiate. Moving a graph up or down never changes its slope.

## Related concepts

- [Product rule](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-5)
- [Quotient rule](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-6)
- [Chain rule](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-7)
- [Derivative](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-1)

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