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Product Rule

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

Product Rule

For any function $F (x) = f (x) \cdot g (x)$

$\begin{aligned} F^{'} (x) & = g (x) \cdot f^{'} (x) + f (x) \cdot g^{'} (x) \\ F^{'} & = g \cdot f^{'} + f \cdot g^{'} \\ F^{'} (x) & = f^{'} (x) \cdot g (x) + f (x) \cdot g^{'} (x) \end{aligned}$

Example: $f (x) = x \sin x$

$\begin{aligned} f^{'} (x) & = \sin x (1) + x \cdot (\cos x) \\ = \sin x + x \cdot \cos x \end{aligned}$

3-function product rule: For any function $F (x) = f (x) \cdot g (x) \cdot h (x)$ ,

$\begin{aligned} F^{'} (x) & = f^{'} (x) \cdot g (x) \cdot h (x) + f (x) \cdot g^{'} (x) \cdot h (x) \\ + f (x) \cdot g (x) \cdot h^{'} (x) \end{aligned}$

Related Glossary Terms

Derivative

Index

Chapter 4 - The Product Rule

Chapter 4 - The Quotient Rule

Chapter 4 - The Chain Rule

Chapter 4 - FFR - Implicit Differentiation

Chapter 5 - Powers and Logarithms

Chapter 5 - Inverse Functions

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.

Product rule
The product rule differentiates a product of two differentiable functions. It gives (fg)′=f′g+fg′, changing one factor at a time and adding the results. For example, the derivative of x sin x is sin x+x cos x. The product of the two derivatives is generally not the derivative of the product.
Read more: 2.8 · Product rule →

Product rule

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