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

The Quotient 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 quotients

How do I differentiate one function divided by another?

  • The derivative of the quotient of two functions can be found using the quotient rule

  • The quotient rule states that

    • If h open parentheses x close parentheses equals fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction,

    • then h to the power of apostrophe open parentheses x close parentheses equals fraction numerator f to the power of apostrophe open parentheses x close parentheses times g open parentheses x close parentheses space minus space f open parentheses x close parentheses times g to the power of apostrophe open parentheses x close parentheses over denominator open parentheses g open parentheses x close parentheses close parentheses squared end fraction

  • This is also commonly written as

    • If y equals u over v,

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

    • Or in a more concise form: y to the power of apostrophe equals fraction numerator u to the power of apostrophe v space minus space u v to the power of apostrophe over denominator v squared end fraction

  • Any quotient rule problem can alternatively be solved as a product rule problem by writing h open parentheses x close parentheses equals fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction as h open parentheses x close parentheses equals f open parentheses x close parentheses times open parentheses g open parentheses x close parentheses close parentheses to the power of negative 1 end exponent

    • The product rule and the chain rule can then be applied

      • h apostrophe open parentheses x close parentheses equals f apostrophe open parentheses x close parentheses times open parentheses g open parentheses x close parentheses close parentheses to the power of negative 1 end exponent plus f open parentheses x close parentheses times open parentheses negative g apostrophe open parentheses x close parentheses open parentheses g open parentheses x close parentheses close parentheses to the power of negative 2 end exponent close parentheses

    • This can be rewritten as

      • h to the power of apostrophe open parentheses x close parentheses equals fraction numerator f apostrophe open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction minus fraction numerator space f open parentheses x close parentheses times g to the power of apostrophe open parentheses x close parentheses over denominator open parentheses g open parentheses x close parentheses close parentheses squared end fraction

    • Multiply the first term by fraction numerator g open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction to get the formula for the quotient rule

      • h to the power of apostrophe open parentheses x close parentheses equals fraction numerator f to the power of apostrophe open parentheses x close parentheses times g open parentheses x close parentheses space minus space f open parentheses x close parentheses times g to the power of apostrophe open parentheses x close parentheses over denominator open parentheses g open parentheses x close parentheses close parentheses squared end fraction

    • However, it is quicker and safer to just use the quotient rule

Examiner Tips and Tricks

Notice that the numerator is the same as the product rule but with a subtraction instead (provided you write the u to the power of apostrophe v term first!).

Product rule: If y equals u times v, then y to the power of apostrophe equals u to the power of apostrophe v space plus space u v to the power of apostrophe.

  • 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 quotient 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 fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction. Find space h apostrophe open parentheses 2 close parentheses. Show the work that leads to your answer.

Answer:

Apply the quotient rule to the function space h

space h apostrophe open parentheses x close parentheses equals fraction numerator f apostrophe open parentheses x close parentheses times g open parentheses x close parentheses minus f open parentheses x close parentheses times g apostrophe open parentheses x close parentheses over denominator open parentheses g open parentheses x close parentheses close parentheses squared end fraction

Substitute x equals 2

space h apostrophe open parentheses 2 close parentheses equals fraction numerator f apostrophe open parentheses 2 close parentheses times g open parentheses 2 close parentheses minus f open parentheses 2 close parentheses times g apostrophe open parentheses 2 close parentheses over denominator open parentheses g open parentheses 2 close parentheses close parentheses squared end fraction

Use the values in the table

space h apostrophe open parentheses 2 close parentheses equals fraction numerator open parentheses negative 1 close parentheses times 5 minus 3 times 4 over denominator open parentheses 5 close parentheses squared end fraction

space h apostrophe open parentheses 2 close parentheses equals negative 17 over 25

Worked Example

Find the derivative of the following functions.

(a) f open parentheses x close parentheses equals fraction numerator 4 x plus 3 over denominator 2 x plus 5 end fraction

(b) g open parentheses x close parentheses equals fraction numerator sin space 3 x over denominator e to the power of 4 x end exponent end fraction

Answer:

(a)

Assign u and v to each function

u equals 4 x plus 3

v equals 2 x plus 5

Find the derivatives of u and v

u to the power of apostrophe equals 4

v to the power of apostrophe equals 2

Apply the quotient rule, y to the power of apostrophe equals fraction numerator u to the power of apostrophe v space minus space u v to the power of apostrophe over denominator v squared end fraction

y to the power of apostrophe equals fraction numerator 4 open parentheses 2 x plus 5 close parentheses minus open parentheses 4 x plus 3 close parentheses times 2 over denominator open parentheses 2 x plus 5 close parentheses squared end fraction

Simplify

y to the power of apostrophe equals fraction numerator open parentheses 8 x plus 20 close parentheses minus open parentheses 8 x plus 6 close parentheses over denominator open parentheses 2 x plus 5 close parentheses squared end fraction

f to the power of apostrophe open parentheses x close parentheses equals 14 over open parentheses 2 x plus 5 close parentheses squared

(b)

Method 1 - Quotient rule

Assign u and v to each function

u equals sin space 3 x

v equals e to the power of 4 x end exponent

Find the derivatives of u and v

u to the power of apostrophe equals 3 cos space 3 x

v to the power of apostrophe equals 4 e to the power of 4 x end exponent

Apply the quotient rule, y to the power of apostrophe equals fraction numerator u to the power of apostrophe v space minus space u v to the power of apostrophe over denominator v squared end fraction

y to the power of apostrophe equals fraction numerator 3 cos space 3 x times e to the power of 4 x end exponent space minus space sin space 3 x times 4 e to the power of 4 x end exponent over denominator open parentheses e to the power of 4 x end exponent close parentheses squared end fraction

In the numerator, the e to the power of 4 x end exponent term can be factored out

y to the power of apostrophe equals fraction numerator e to the power of 4 x end exponent open parentheses 3 cos space 3 x space minus space 4 sin space 3 x close parentheses over denominator open parentheses e to the power of 4 x end exponent close parentheses squared end fraction

Simplify the powers of e to the power of 4 x end exponent

g to the power of apostrophe open parentheses x close parentheses equals fraction numerator 3 cos space 3 x space minus space 4 sin space 3 x over denominator e to the power of 4 x end exponent end fraction

Method 2 - Product rule

Rewrite as a product

space g open parentheses x close parentheses equals sin 3 x times e to the power of negative 4 x end exponent

Assign u and v to each function

u equals sin space 3 x

v equals e to the power of negative 4 x end exponent

Find the derivatives of u and v

u to the power of apostrophe equals 3 cos space 3 x

v to the power of apostrophe equals negative 4 e to the power of negative 4 x end exponent

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 3 cos 3 x times e to the power of negative 4 x end exponent plus sin 3 x times open parentheses negative 4 e to the power of negative 4 x end exponent close parentheses

Simplify

g to the power of apostrophe open parentheses x close parentheses equals open parentheses 3 cos space 3 x space minus space 4 sin space 3 x close parentheses e to the power of negative 4 x end exponent

Examiner Tips and Tricks

In part (b) of the worked example above, both methods give equivalent answers. In your exam, you are not usually required to simplify your answers, so you would not need to cancel the common term in the fraction in method 1 or factor the common term in method 2.

However, method 1 is definitely safer, as you are less likely to make a sign error. The only time a student should use method 2 is if they forget the formula for the quotient rule.

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Author
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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.