﻿ How to Use the Product Rule for Derivatives. Visual Explanation with color coded examples

# How to Use the Product Rule for Derivatives

### Quick Overview

• $$\displaystyle \frac d {dx}\left(f(x)\cdot g(x)\right) = f'(x)g(x) + f(x)g'(x)$$
• The examples and questions in this lesson assume you know the Basic Derivatives from earlier lessons. ### Explanation

When differentiating a product, each factor is differentiated, but one at a time. You can think of this as "each factor gets a turn at being the derivative." This pattern holds true, even when there are more than two factors in the product, as shown below.

$$\frac d {dx}\left(f(x)\cdot g(x) \cdot h(x)\right) = \blue{f'(x)}\cdot g(x)\cdot h(x) + f(x)\cdot \blue{g'(x)}\cdot h(x) + f(x)\cdot g(x)\cdot\blue{h'(x)}$$

### Examples

##### Example 1---Two Factors

Suppose $$f(x) = x^2\sin 3x$$. Find $$f'(x)$$.

Step 1

Identify the factors in the function.

$$f(x) = \blue{x^2}\cdot \red{\sin 3x}$$

Step 2

Use the product rule to differentiate the function.

$$f'(x) = \blue{2x}\sin 3x + x^2\red{(3\cos 3x)} = 2x\sin 3x + 3x^2\cos 3x$$

$$f'(x) = 2x\sin 3x + 3x^2\cos 3x$$

##### Example 2---Three Factors

Suppose $$f(x) = 4x^3\,e^{-2x}\cos 6x$$. Find $$f'(x)$$

Step 1

Identify the factors that make up the function.

$$f(x) = 4x^3\blue{e^{-2x}}\red{\cos 6x}$$

Step 2

Differentiate using the product rule. The parts in $$\blue{blue}$$ are the derivatives of the individual factors.

\begin{align*}% f'(x) & = \big[\blue{12x^2}\,e^{-2x}\cos 6x\big] + \big[4x^3\blue{(-2e^{-2x})}\cos 6x\big] + \big[4x^3\,e^{-2x}\blue{(-6\sin 6x)}\big]\\[6pt] & = 12x^2\,e^{-2x}\cos 6x - 8x^3e^{-2x}\cos 6x - 24x^3\,e^{-2x}\sin 6x \end{align*}

Step 3

(Optional) Write the derivative in a factored form.

\begin{align*} f'(x) & = 12x^2\,\blue{e^{-2x}}\cos 6x - 8x^3\blue{e^{-2x}}\cos 6x - 24x^3\,\blue{e^{-2x}}\sin 6x\\[6pt] & = \blue{e^{-2x}}\left(12\red{x^2}\cos 6x - 8\red{x^3}\cos 6x - 24\red{x^3}\sin 6x\right)\\[6pt] & = \red{x^2}e^{-2x}\left(12\cos 6x - 8x\cos 6x - 24x\sin 6x\right)\\[6pt] & = 4x^2e^{-2x}\left(3\cos 6x - 2x\cos 6x - 6x\sin 6x\right) \end{align*}

$$f'(x) = 4x^2e^{-2x}\left(3\cos 6x - 2x\cos 6x - 6x\sin 6x\right)$$.
{"ft_lang_label":"__label__en","ft_lang_prob":0.5797065,"math_prob":0.9999912,"size":2034,"snap":"2020-45-2020-50","text_gpt3_token_len":904,"char_repetition_ratio":0.16009852,"word_repetition_ratio":0.043165468,"special_character_ratio":0.4346116,"punctuation_ratio":0.05773672,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99999297,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,5,null,5,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-25T05:00:21Z\",\"WARC-Record-ID\":\"<urn:uuid:a6611b63-b477-496f-9e8e-6ee847677296>\",\"Content-Length\":\"33929\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:58b6bd69-0b0b-4486-94e8-09f383186808>\",\"WARC-Concurrent-To\":\"<urn:uuid:118c1a75-59cb-446d-94a1-f9d63af4aa00>\",\"WARC-IP-Address\":\"104.24.112.201\",\"WARC-Target-URI\":\"https://www.mathwarehouse.com/calculus/derivatives/how-to-use-the-product-rule.php\",\"WARC-Payload-Digest\":\"sha1:MNV5U67KV6CD6DUG7R5T5SJODAC5DETK\",\"WARC-Block-Digest\":\"sha1:FLVV2XWIMENSGMK72F6LCQR73SLHA77G\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141181179.12_warc_CC-MAIN-20201125041943-20201125071943-00283.warc.gz\"}"}