The Product Rule (College Board AP® Calculus AB)

Study Guide

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Written by: Jamie Wood

Reviewed by: Dan Finlay

Derivatives of products

How do I differentiate the product of two functions?

The derivative of the product of two functions can be found by using the product rule
The product rule states that
- If $f (x) = g (x) \cdot h (x)$ ,
- then $f^{'} (x) = g^{'} (x) \cdot h (x) + g (x) \cdot h^{'} (x)$
This is also commonly written as
- If $y = u \cdot v$ ,
- then $\frac{d y}{d x} = \frac{d u}{d x} \cdot v + u \cdot \frac{d v}{d x}$
- Or in a more concise form: $y^{'} = u^{'} v + u v^{'}$

Examiner Tips and Tricks

Don't confuse the product of two functions with a composite function
- The product of two functions like $f (x) \cdot g (x)$ is two functions multiplied together
- A composite function like $f (g (x))$ is a function of a function
- For how to differentiate composite functions, see the "Chain rule" study guide

Worked Example

Find the derivative of the following functions.

(a) $f (x) = e^{2 x} (x^{5} + 3 x)$

Answer:

Assign $u$ and $v$ to each function

$u = e^{2 x}$

$v = x^{5} + 3 x$

Find the derivatives of $u$ and $v$

$u^{'} = 2 e^{2 x}$

$v^{'} = 5 x^{4} + 3$

Apply the product rule, $y^{'} = u^{'} v + u v^{'}$

$y^{'} = 2 e^{2 x} (x^{5} + 3 x) + e^{2 x} (5 x^{4} + 3)$

If you have used a different notation such as $y^{'}$ when working, you should write your final answer in a format that matches how the question was asked

$f^{'} (x) = 2 e^{2 x} (x^{5} + 3 x) + e^{2 x} (5 x^{4} + 3)$

This answer can also be factorized

$f^{'} (x) = e^{2 x} (2 x^{5} + 5 x^{4} + 6 x + 3)$

(b) $g (x) = \ln 3 x \cdot \sin 2 x$

Answer:

Assign $u$ and $v$ to each function

$u = \ln 3 x$

$v = \sin 2 x$

Find the derivatives of $u$ and $v$

$u^{'} = \frac{1}{x}$

$v^{'} = 2 \cos 2 x$

Apply the product rule, $y^{'} = u^{'} v + u v^{'}$

$y^{'} = \frac{1}{x} \cdot \sin 2 x + \ln 3 x \cdot 2 \cos 2 x$

Simplify

$g^{'} (x) = \frac{\sin 2 x}{x} + 2 \ln (3 x) \cos (2 x)$

Last updated: 5 August 2024

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