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

The Quotient Rule (College Board AP® Calculus AB)

Study Guide

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

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

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

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.

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

Answer:

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 equals 14 over open parentheses 2 x plus 5 close parentheses squared

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

(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:

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

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    Previous:The Product RuleNext:Derivatives of Tangent and Reciprocal Trig Functions
    Author
    Reviewer
    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths

    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

    Author: Dan Finlay

    Expertise: Maths 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.