Calculus — Derivatives⏱ 5 min read

Power Rule

d/dx[xⁿ] = nxⁿ⁻¹

What is the Power Rule?

The power rule is the single most-used rule in differential calculus. To differentiate any term with a variable raised to a power, multiply the term by that exponent, then reduce the exponent by one. It works for positive, negative, and fractional exponents alike.

Two special cases fall directly out of the power rule: the derivative of a constant is always 0 (since a constant doesn't change), and the derivative of x itself is always 1 (since x = x¹, and the rule gives 1·x⁰ = 1).

What Each Variable Means

d/dx

Derivative operatorThe rate of change of the function with respect to x.

xⁿ

The functionx raised to a power — the function being differentiated.

ExponentCan be any real number: positive, negative, zero, or a fraction.

When to Use It

Differentiating any single power term, like x³, x⁻², or √x = x^(1/2)
As a building block inside more complex derivatives, combined with the product, quotient, and chain rules
Quickly checking the slope of a polynomial curve at any point

Step-by-Step Examples

Example 1: A simple power

Problem: Differentiate y = x³

Multiply by the exponent, reduce it by 1

Apply the rule directly.

d/dx[x³] = 3x²

✓

Answer: dy/dx = 3x²

Example 2: A negative exponent

Problem: Differentiate y = x⁻²

Apply the power rule as normal

The rule works the same way regardless of the exponent's sign.

d/dx[x⁻²] = -2x⁻³

Rewrite with a positive exponent if preferred

Optional, for a cleaner final form.

-2x⁻³ = -2/x³

✓

Answer: dy/dx = -2/x³

Common Mistakes

Mistake: Forgetting to reduce the exponent by one after multiplying.
Fix: The power rule has two steps, not one: multiply by n, then also change the exponent to n−1. Doing only the multiplication is a common shortcut error.
Mistake: Applying the power rule to a base other than x, like 2ˣ.
Fix: The power rule only applies when the variable is the base and the exponent is a constant. When the variable is in the exponent instead, use the exponential derivative rule (d/dx[aˣ] = aˣ ln a) instead.