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Power rule

This original textbook reading covers part of the AP topic. Follow the AP course outline for complete topic instruction and practice.

Power rule

$\frac{d}{d x} x^{n} = n \cdot x^{n - 1}$ . This is a shortcut to finding the derivative of a polynomial function.

Example: $f (x) = x^{4} - x^{3} + 3 x^{2} + x + 8$

$\begin{aligned} f^{'} (x) & = 4 \cdot x^{4 - 1} - 3 \cdot x^{3 - 1} + 3 \cdot 2 \cdot x^{2 - 1} - 1 \cdot x^{1 - 1} + 0 \\ = 4 \cdot x^{3} - 3 \cdot x^{2} + 6 \cdot x - 1 \\ = 4 x^{3} - 3 x^{2} + 6 x - 1 \end{aligned}$

Related Glossary Terms

Anti-Power Rule, Derivative

Index

Chapter 3 - Differentiability

Chapter 3 - FFR - Derivative Concepts

Chapter 4 - The Product Rule

Chapter 4 - The Chain Rule

Glossary and deeper reading · 1 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

Power rule
The derivative power rule gives d(xⁿ)/dx=n xⁿ⁻¹ wherever the real power function is differentiable. Constant multiples remain as constant factors. For example, the derivative of 3x⁴ is 12x³. A composite power such as (x²+1)³ also requires the chain rule.
Read more: 2.5 · Power rule →

Power rule

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