The Product Rule | AP Calc AB Unit 2 Study Guide

The Core Idea: The Product Rule

The Product Rule addresses a fundamental question in differentiation: how do we find the derivative of a function that is created by multiplying two other differentiable functions together? A common but incorrect assumption is that the derivative of a product is simply the product of the derivatives. The Product Rule provides the correct method for this calculation.

This rule is essential for expanding the types of functions we can differentiate. It allows us to handle complex functions by breaking them down into simpler, multiplied parts. The core idea is that the rate of change of a product, $f (x) g (x)$ , depends on both the values of the functions and their respective rates of change. The rule combines these four pieces of information— $f (x)$ , $g (x)$ , $f^{'} (x)$ , and $g^{'} (x)$ —in a specific way to find the overall derivative.

Key Formulas/Rules/Theorems

The Product Rule provides the formula for differentiating the product of two differentiable functions.

Let $h (x) = f (x) g (x)$ , where $f$ and $g$ are both differentiable functions. The derivative of $h (x)$ is given by:

$h^{'} (x) = f^{'} (x) g (x) + f (x) g^{'} (x)$

In words, the derivative of a product of two functions is: "the derivative of the first function times the second function, plus the first function times the derivative of the second function."

Using Leibniz notation, the rule is expressed as:

$\frac{d}{d x} [f (x) g (x)] = \frac{d}{d x} [f (x)] \cdot g (x) + f (x) \cdot \frac{d}{d x} [g (x)]$

Understanding The Product of Multiple Functions

The Essential Knowledge for this topic specifies that the Product Rule can be used to find the derivative of a product of "two or more" differentiable functions. To find the derivative of a product of three functions, such as $h (x) = f (x) g (x) k (x)$ , we can apply the Product Rule iteratively.

First, group the functions into two parts, for example, $[f (x) g (x)]$ and $[k (x)] F or m u l a [2] h^{'} (x) = \frac{d}{d x} [f (x) g (x)] \cdot k (x) + [f (x) g (x)] \cdot k^{'} (x) F or m u l a [3] \frac{d}{d x} [f (x) g (x)] = f^{'} (x) g (x) + f (x) g^{'} (x) F or m u l a [4] h^{'} (x) = [f^{'} (x) g (x) + f (x) g^{'} (x)] \cdot k (x) + f (x) g (x) k^{'} (x) F or m u l a [5] h^{'} (x) = f^{'} (x) g (x) k (x) + f (x) g^{'} (x) k (x) + f (x) g (x) k^{'} (x) F or m u l a [6] h^{'} (x) = (8 x) (x^{3} + 6 x) + (4 x^{2} - 1) (3 x^{2} + 6) F or m u l a [7] h^{'} (x) = (8 x^{4} + 48 x^{2}) + (12 x^{4} + 21 x^{2} - 6) F or m u l a [8] h^{'} (x) = 20 x^{4} + 69 x^{2} - 6 F or m u l a [9] h^{'} (x) = f^{'} (x) g (x) + f (x) g^{'} (x) F or m u l a [10] h^{'} (3) = f^{'} (3) g (3) + f (3) g^{'} (3) F or m u l a [11] h^{'} (3) = (- 4) (5) + (2) (6) F or m u l a [12] h^{'} (3) = - 20 + 12 F or m u l a [13] h^{'} (3) = - 8$ $

Using Your Calculator

The Product Rule is an analytical rule for finding a derivative function by hand. A calculator cannot perform the symbolic steps of applying the rule. However, it is an excellent tool for verifying your answer at a specific point.

Suppose you used the Product Rule to find that the derivative of $h (x) = (4 x^{2} - 1) (x^{3} + 6 x)$ is $h^{'} (x) = 20 x^{4} + 69 x^{2} - 6$ , as in Example 1. You can check your work at a point, say $x = 2$ .

1. Analytical Calculation (By Hand):

$h^{'} (2) = 20 (2)^{4} + 69 (2)^{2} - 6 = 20 (16) + 69 (4) - 6 = 320 + 276 - 6 = 590$

2. Calculator Verification:

Use the numerical derivative function (often nDeriv or a $d / d x$ template).
Enter the original function $h (x)$ and the point $x = 2$ .
On a TI-84 style calculator, the input would look like: nDeriv((4X^2-1)(X^3+6X), X, 2)
The calculator will return $590$ , confirming that your analytical derivative function is likely correct.

This method does not find the derivative function for you, but it provides a reliable check for your calculations at any single point.

AP Exam Quick Hit

Common Question Types

Direct Symbolic Differentiation: You will be given a function that is an explicit product of two other functions (e.g., $k (x) = x^{2} cos (x)$ ) and asked to find $k^{'} (x)$ . This tests your direct application of the formula.
Applying from a Table: You will be given a table of values for $f (c)$ , $f^{'} (c)$ , $g (c)$ , and $g^{'} (c)$ and asked to find the derivative of $f (x) g (x)$ at $x = c$ . This tests whether you have memorized the formula and can apply it correctly with given numbers.
Applying from Graphs: You will be shown the graphs of two functions, $f$ and $g$ , and asked to find the derivative of their product at a point $x = c$ . This requires you to find the function values ( $y$ -coordinates) and the derivative values (slopes of the tangent lines) from the graphs before substituting them into the Product Rule formula.

Common Mistakes

The "Fake" Product Rule: The most frequent error is assuming the derivative of a product is the product of the derivatives: $\frac{d}{d x} [f (x) g (x)] = f^{'} (x) g^{'} (x)$ . This is fundamentally incorrect and will always result in a wrong answer.
Algebraic and Sign Errors: After correctly applying the rule, errors often occur during the simplification step. Be very careful with distribution, especially when negative signs are involved (e.g., distributing a term like $(x - 5)$ ).
Mixing up the Parts: A simple memory error where a student swaps a function with its derivative. For example, writing $f (x) g (x) + f^{'} (x) g^{'} (x)$ instead of the correct $f^{'} (x) g (x) + f (x) g^{'} (x)$ .
Incomplete Application: Forgetting one of the two terms in the sum. For instance, only calculating $f^{'} (x) g (x)$ and stopping, forgetting to add the $+ f (x) g^{'} (x)$ part.

The Product Rule - AP Calculus AB Study Guide