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The Product Rule

Hannah Hill

Hannah Hill

6 min read

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Study Guide Overview

This study guide covers the product rule for finding derivatives. It includes the formula, a step-by-step walkthrough, practice problems, and common mistakes. It emphasizes the importance of this rule for the AP Calculus exam, providing practice multiple choice and free-response questions with answers and a scoring rubric.

The Product Rule 🚀

Welcome back! Let's master the product rule, a crucial tool for finding derivatives of functions multiplied together. This is a key skill that you'll see throughout the AP Calculus exam.


🏁 Product Rule Definition

The product rule helps us find the derivative of two functions multiplied together. It's not as simple as just multiplying the derivatives!


Formula: ddx(f(x)g(x))=f(x)g′(x)+g(x)f′(x)\frac{d}{dx}(\textcolor{red}{f(x)}\textcolor{green}{g(x)})= \textcolor{red}{f(x)} \textcolor{blue}{g'(x)} + \textcolor{green}{g(x)} \textcolor{pink}{f'(x)}


Memory Aid

Mnemonic: "First d second plus second d first"

  • First function times the derivative of the second
  • Plus the second function times the derivative of the first

Key Concept

The derivative of a product is NOT the product of the derivatives.


✏️ Product Rule: Walkthrough

Let's break it down with an example:

f(x)=sin⁡(x)(x2+2x)f(x) = \sin(x)(x^2 + 2x)

Applying the product rule:

f′(x)=sin⁡(x)ddx(x2+2x)+(x2+2x)ddx(sin⁡(x))f'(x) = \sin(x) \frac{d}{dx}(x^2 + 2x) + (x^2+2x) \frac{d}{dx}(\sin(x))

f′(x)=sin⁡(x)(2x+2)+(x2+2x)cos⁡(x)f'(x) = \sin(x)(2x+2) + (x^2+2x)\cos(x)


Common Mistake

A common mistake is to incorrectly calculate the derivative as cos(x)(2x+2)cos(x)(2x+2). This is incorrect! Always use the product rule.


Here's a visual to show the difference between the correct and incorrect derivative:


Previous Topic - Derivatives of cos x, sinx, e^x, and ln xNext Topic - The Quotient Rule

Question 1 of 8

If f(x)=xsin⁡(x)f(x) = x \sin(x), what is f′(x)f'(x)?

cos⁡(x)\cos(x)

xcos⁡(x)x \cos(x)

sin⁡(x)+xcos⁡(x)\sin(x) + x \cos(x)

xcos⁡(x)+sin⁡(x)x \cos(x) + \sin(x)