The Power Rule: Interactive Derivative Guide
The power rule says the derivative of x to the n is n times x to the (n minus 1): you bring the exponent down as a multiplier and lower the exponent by one. Constant multiples are kept and constants differentiate to zero.
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:
What is the power rule for derivatives?
Power rule worked example: differentiating a polynomial
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:
Another worked example
Differentiate f(x) = 5x to the 4th minus 3x. Step 1: for 5x to the 4th, bring down 4 and lower the exponent to get 20x to the 3rd. Step 2: for minus 3x, the derivative of x is 1, giving minus 3. Step 3: combine to get f-prime(x) = 20x to the 3rd minus 3.
Aligned with the College Board AP Calculus CED — Unit 2 (Differentiation: Definition and Fundamental Properties), Topic 2.5, with the constant-multiple and sum rules of Topic 2.6. College Board ↗
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