The Power Rule

Introduction

The power rule is one of the most fundamental and useful differentiation rules in calculus.

The Rule

If $f(x) = x^n$ where $n$ is any real number, then:

$f'(x) = nx^{n-1}$

How to Use It

1. Multiply by the exponent
2. Subtract 1 from the exponent

Examples

Example 1: Basic Application

$\frac{d}{dx}(x^5) = 5x^4$

Example 2: With Coefficients

$\frac{d}{dx}(3x^7) = 3 \cdot 7x^6 = 21x^6$

Example 3: Negative Exponents

$\frac{d}{dx}(x^{-2}) = -2x^{-3} = \frac{-2}{x^3}$

Combined with Other Rules

The power rule works great with the sum rule:

$\frac{d}{dx}(x^3 + 2x^2 - 5x + 7) = 3x^2 + 4x - 5$