The Power Rule: Interactive Derivative Guide

The Power Rule is one of the most fundamental shortcuts in calculus. It allows you to find the derivative of any polynomial term without computing the limit definition manually.

To differentiate using the Power Rule:

1. Bring the current exponent n down to the front to multiply the term.

2. Decrease the power of the exponent by exactly 1.

3. Apply this rule term-by-term for polynomial sums, combining with constant multipliers:

\frac{d}{dx}[c x^n] = c \cdot n x^{n-1}

Formula

\frac{d}{dx}[x^n] = n x^{n-1}

Easy Example Problem

Calculate the derivative of f(x) = 4x³ + 2x² - 5x + 7.

1. Apply the power rule term-by-term:

- For 4x³: 3 ➔ 4 * 3x² = 12x²

- For 2x²: 2 ➔ 2 * 2x¹ = 4x

- For -5x: 1 ➔ -5 * 1x⁰ = -5

- For 7 (constant): 0 ➔ 0

2. Add the results together:

f'(x) = 12x^2 + 4x - 5

Interactive Preview

Interactive simulator is loading...

Engaged with the simulation?

Practice actual exam problems and master calculus guaranteed. Get a 5 on the AP Calculus exam or get an A in the calculus class, or we'll pay you $100.

Accept the Mission