AP®MathsCollege BoardCalculus BCStudy GuidesUnit 2: Differentiation Definition & Fundamental PropertiesFundamental Properties of DifferentiationThe Product Rule

The Product Rule (College Board AP® Calculus BC): Study Guide

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Updated on

College Board Calculus BCCollege Board Calculus AB

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 h open parentheses x close parentheses equals f open parentheses x close parentheses times g open parentheses x close parentheses,

    • then h to the power of apostrophe open parentheses x close parentheses equals f to the power of apostrophe open parentheses x close parentheses times g open parentheses x close parentheses space plus space f open parentheses x close parentheses times g to the power of apostrophe open parentheses x close parentheses

  • This is also commonly written as

    • If y equals u times v,

    • then fraction numerator d y over denominator d x end fraction equals fraction numerator d u over denominator d x end fraction times v space plus space u times fraction numerator d v over denominator d x end fraction

    • Or in a more concise form: y to the power of apostrophe equals u to the power of apostrophe v space plus space u v to the power of apostrophe

Examiner Tips and Tricks

Don't confuse the product of two functions with a composite function

  • The product of two functions like f open parentheses x close parentheses times g open parentheses x close parentheses is two functions multiplied together

  • A composite function like f open parentheses g open parentheses x close parentheses close parentheses is a function of a function

  • For how to differentiate composite functions, see the "Chain rule" study guide

You do not need to know the proof of the formula for the product rule, as it is beyond the scope of the course.

  • You might be given a table of values for two functions and their derivatives at a point

  • These questions test whether you fully understand the formula for the product rule

Worked Example

space x

2

space f open parentheses x close parentheses

3

space f apostrophe open parentheses x close parentheses

-1

space g open parentheses x close parentheses

5

space g apostrophe open parentheses x close parentheses

4

The functions space f and space g are differentiable. The table shown gives values of the functions and their first derivatives at x equals 2.

Let space h be the function defined by space h open parentheses x close parentheses equals f open parentheses x close parentheses times g open parentheses x close parentheses. Find space h apostrophe open parentheses 2 close parentheses. Show the work that leads to your answer.

Answer:

Apply the product rule to the function space h

space h apostrophe open parentheses x close parentheses equals f apostrophe open parentheses x close parentheses times g open parentheses x close parentheses plus f open parentheses x close parentheses times g apostrophe open parentheses x close parentheses

Substitute x equals 2

space h apostrophe open parentheses 2 close parentheses equals f apostrophe open parentheses 2 close parentheses times g open parentheses 2 close parentheses plus f open parentheses 2 close parentheses times g apostrophe open parentheses 2 close parentheses

Use the values in the table

space h apostrophe open parentheses 2 close parentheses equals open parentheses negative 1 close parentheses times 5 plus 3 times 4

space h apostrophe open parentheses 2 close parentheses equals 7

Worked Example

Find the derivative of the following functions.

(a) f open parentheses x close parentheses equals e to the power of 2 x end exponent open parentheses x to the power of 5 plus 3 x close parentheses

(b) g open parentheses x close parentheses equals ln space 3 x times sin space 2 x

Answer:

(a)

Assign u and v to each function

u equals e to the power of 2 x end exponent

v equals x to the power of 5 plus 3 x

Find the derivatives of u and v

u to the power of apostrophe equals 2 e to the power of 2 x end exponent

v to the power of apostrophe equals 5 x to the power of 4 plus 3

Apply the product rule, y to the power of apostrophe equals u to the power of apostrophe v space plus space u v to the power of apostrophe

y to the power of apostrophe equals 2 e to the power of 2 x end exponent open parentheses x to the power of 5 plus 3 x close parentheses plus e to the power of 2 x end exponent open parentheses 5 x to the power of 4 plus 3 close parentheses

If you have used a different notation such as y to the power of apostrophe when working, you should write your final answer in a format that matches how the question was asked

f to the power of apostrophe open parentheses x close parentheses equals 2 e to the power of 2 x end exponent open parentheses x to the power of 5 plus 3 x close parentheses plus e to the power of 2 x end exponent open parentheses 5 x to the power of 4 plus 3 close parentheses

This answer can also be factored

f to the power of apostrophe open parentheses x close parentheses equals e to the power of 2 x end exponent open parentheses 2 x to the power of 5 plus 5 x to the power of 4 plus 6 x plus 3 close parentheses

(b)

Assign u and v to each function

u equals ln space 3 x

v equals sin space 2 x

Find the derivatives of u and v

u to the power of apostrophe equals 1 over x

v to the power of apostrophe equals 2 cos space 2 x

Apply the product rule, y to the power of apostrophe equals u to the power of apostrophe v space plus space u v to the power of apostrophe

y to the power of apostrophe equals 1 over x times sin space 2 x space plus space ln space 3 x times 2 cos space 2 x

Simplify

g to the power of apostrophe open parentheses x close parentheses equals fraction numerator sin space 2 x over denominator x end fraction plus 2 ln open parentheses 3 x close parentheses cos open parentheses 2 x close parentheses

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

I would just like to say a massive thank you for putting together such a brilliant, easy to use website.I really think using this site helped me secure my top gradesin science and maths. You really did save my exams! Thank you.

Beth
IGCSE Student

This website is soooo useful and I can’t ever thank you enough for organising questions by topic like this. Furthermore, the name of the website could not have been more appropriate as it literally did SAVE MY EXAMS!

Fathima
A Level Student

Incredible! SO worth my money, the revision notes have everything I need to know and are so easy to understand. I actually enjoy revising! It makes me feel a lot more confident for my GCSEs in a few months.

Kate
GCSE Student

Absolutely brilliant, both my girls used it for A levels and GCSE. It's saves on paper copies, also beneficial exam questions ranked from easy to hard. It's removed a lot of stress from the exams.

Sameera
Parent

Just to say that your resources are the best I have seen and I have been teaching chemistry at different levels for about 40 years

Mark
Chemistry Teacher

Excellent

Our rating represented in Trustpilot stars
Read more reviews
Test yourself
Previous:Derivatives of Sine and Cosine FunctionsNext:The Quotient Rule
Author
Reviewer
Jamie Wood

Author: Jamie Wood

Expertise: Maths Content Creator

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

Dan Finlay

Reviewer: Dan Finlay

Expertise: Maths Subject Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.