Power rule
Also known as: power rule for derivatives
The power rule is the differentiation shortcut stating that the derivative of xⁿ is n·xⁿ⁻¹: bring the exponent down as a coefficient, then reduce the exponent by one. It works for any real number exponent.
The power rule is the most-used shortcut in differential calculus: for any real number n, the derivative of xⁿ is n·xⁿ⁻¹. In words, multiply by the old exponent, then subtract one from it. The derivative of x³ is 3x², the derivative of x¹⁰ is 10x⁹, and the derivative of x itself is 1.
The rule's reach comes from rewriting. Roots and reciprocals become powers first: √x = x^(1/2), so its derivative is (1/2)x^(−1/2); 1/x = x^(−1), so its derivative is −x^(−2). Combined with the constant multiple and sum rules, the power rule differentiates any polynomial term by term — the derivative of 4x³ − 7x + 2 is 12x² − 7. One caution: the rule applies to a variable raised to a constant power, not a constant raised to a variable power — the derivative of 2ˣ requires exponential rules instead.
In algebra courses, "power rule" also names the exponent law (xᵃ)ᵇ = xᵃᵇ — raising a power to a power multiplies the exponents, so (x²)³ = x⁶. The two rules live in different topics but reward the same habit: rewrite expressions into clean power form before applying the rule.
The AP Calculus AB exam uses the derivative power rule constantly, from basic derivative questions through optimization and related rates, while the ASVAB and Praxis Core Math test the exponent version as part of rules and operations with powers.
Key takeaways
- Power rule for derivatives: the derivative of xⁿ is n·xⁿ⁻¹, valid for any real exponent n.
- Rewrite roots and reciprocals as powers first: √x = x^(1/2) and 1/x = x^(−1).
- With the sum and constant multiple rules, it differentiates any polynomial term by term.
- It applies to variables raised to constant powers — not to constants raised to variable powers like 2ˣ.
- In algebra, the power rule for exponents says (xᵃ)ᵇ = xᵃᵇ.