Power rule: what it means and how to use it
Differentiate a power of the variable.
Power rule is one of 9 derivatives formulas in the calculus section of this library, and it is used at ap · university level.
Why power rule works
Expanding (x+h) raised to the n gives x to the n, plus n times x to the n−1 times h, plus terms that all carry h² or a higher power. Subtracting x to the n and dividing by h leaves n times x to the n−1 plus terms that still contain an h, and those disappear as h shrinks.
What each symbol means
$n$ is a constant exponent.
Power rule: when it holds
Valid where $x^n$ is differentiable; domain restrictions still apply for fractional or negative powers.
When it stops applying
The rule handles a power of the variable, not a power with the variable upstairs. The derivative of 2 to the x is not x times 2 to the x−1, because there the exponent is what moves; that case needs the exponential rule instead.
Power rule: a worked example
$\frac{d}{dx}x^5=5x^4$.
The mistake to avoid
What people do: Writing the derivative of x to the fifth as 5x to the fifth, keeping the exponent where it was.
Why it goes wrong: The rule asks for two moves, and only one was done: the exponent must come down to the front and also drop by one, so the degree of the answer is wrong.
Do this instead: Do both moves in order. Copy the exponent to the front, then subtract 1 from it, which gives 5x to the fourth.
Power rule: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $n$ is a constant exponent.
- Check the conditions before substituting. Valid where $x^n$ is differentiable; domain restrictions still apply for fractional or negative powers.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Derivatives
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where power rule comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Derivative Rules lesson — the lesson behind this formula: differentiate sums, products, quotients, and compositions.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about power rule
Does the rule cover negative and fractional exponents?
Yes. The derivative of x to the −2 is −2x to the −3, and the derivative of x to the one half is one half times x to the negative one half. Only the allowed inputs change.
What does the rule give when the exponent is 0?
x to the zero is the constant 1, and the rule returns 0 times x to the −1, which is 0 for nonzero x. That matches the fact that a constant never changes.
Why does the exponent drop rather than climb?
Slope is one step simpler than height. A curve of degree n has a slope that behaves like degree n−1, so each derivative trades a factor of x for a plain number out front.
Can I use it on (3x+1) raised to the fourth?
Not on its own. The base is a whole expression rather than a bare x, so you also have to multiply by the derivative of the inside, which is the chain rule.