Back To Course

AP Calculus AB & BC: Help and Review17 chapters | 158 lessons

Watch short & fun videos
**Start Your Free Trial Today**

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will explain how to find the difference quotient of functions with fractions. We will look at the steps involved in this process, and we will apply these steps to an example to help solidify our understanding.

The **difference quotient** for a function *f*(*x*) is given by the formula

[*f*(*x* + *h*) - *f*(*x*)] / *h*

This difference quotient calculates the slope of the secant line through the points (*x*, *f*(*x*)) and (*x* + *h*, *f*(*x* + *h*)), and it is heavily used when dealing with derivatives in calculus.

Finding the difference quotient for a function *f*(*x*) is simply a matter of finding *f*(*x*+*h*) and plugging both *f*(*x*) and *f*(*x*+*h*) into the difference quotient formula, then simplifying.

This all may be old news to you; however, what you may not be familiar with is how to find the difference quotient of a function that has fractions. Thankfully, other than the simplification part, the process of finding the difference quotient for functions with fractions really isn't too much different than it is for functions without fractions.

In fact, the first step in finding the difference quotient for a function with fractions is exactly the same as for functions without fractions. We find *f*(*x* + *h*) and plug both *f*(*x*) and *f*(*x* + *h*) into the formula for the difference quotient. Once we do this, things will look a little different and seem a little trickier.

You see, your difference quotient, which is a fraction itself, will have fractions within it, so you have fractions within fractions. Well, that sounds confusing! Don't worry though! There is a nice trick for eliminating these fractions within the difference quotient, making simplification much easier. All we have to do is find the **common denominator** of the fractions within the difference quotient, and then multiply both the numerator and denominator by that common denominator.

This is okay to do, because it is the same as multiplying by 1. Therefore, we're not changing any values or anything, and what's great is that doing this will eliminate the fractions within the difference quotient, so we can simplify as we normally would.

This might sound like a bit much. I don't know about you, but understanding is always easier for me when I've got a nicely stepped out process and I can see that process used on an actual example, so let's summarize these steps into a nice organized list, and then we'll look at an example to help solidify our understanding of how to do each of the steps.

To find the difference quotient for a function, *f*(*x*), that contains fractions, we use the following steps:

- Find
*f*(*x*+*h*) and plug both*f*(*x*) and*f*(*x*+*h*) into the difference quotient, [*f*(*x*+*h*) -*f*(*x*)] /*h*. - Find the common denominator of all the fractions within the difference quotient and multiply the numerator and denominator of the difference quotient by that common denominator. Quick note: when you find the common denominator, keep it in factored form, and do not multiply it out. This will make simplification easier.
- Simplify the result of step 2.

Okay, that's nicely stepped out, but it will become even more clear when we actually look at applying it to an example, so let's go ahead and do just that.

Consider the function *f*(*x*) = 5 / (*x* - 4). This is a function that is a fraction, so let's use this function to illustrate the steps of finding the difference quotient of a function with fractions.

The first step is no different than finding the difference quotient for any function. We find *f*(*x* + *h*) by plugging in *x* + *h* anywhere we see *x* in the function.

We get that *f*(*x*+*h*) = 5 / (*x* + *h* - 4). Now we simply plug *f*(*x*) and *f*(*x* + *h*) into the difference quotient formula.

So far, so good!

Now comes the fun part! We want to eliminate those fractions within the difference quotient, so we find the common denominator of those fractions. The denominators of the fractions within the quotient are *x* + *h* - 4 and *x* - 4. The easiest way to find a common denominator is to just multiply the denominators together to get

- (
*x*+*h*- 4)(*x*- 4)

As our steps indicate, we should leave this in factored form. Multiplying it out will actually make things more confusing when we multiply the numerator and denominator by this expression. Speaking of which, that's the next step in this process, so let's do just that!

Ah-ha! No more fractions, just as we expected!

Now, we just need to simplify this expression.

We're done!

That wasn't so bad, and we see that finding the difference quotient of functions with fractions is really just a matter of the extra step of finding the common denominator of the fractions within and multiplying the numerator and denominator by that common denominator to eliminate fractions. Other than that, it's business as usual!

To unlock this lesson you must be a Study.com Member.

Create
your account

Already a member? Log In

BackDid you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
9 in chapter 4 of the course:

Back To Course

AP Calculus AB & BC: Help and Review17 chapters | 158 lessons

- Using a Graph to Define Limits 5:24
- Understanding Limits: Using Notation 3:43
- One-Sided Limits and Continuity 4:33
- How to Determine the Limits of Functions 5:15
- Understanding the Properties of Limits 4:29
- Squeeze Theorem: Definition and Examples 5:49
- Graphs and Limits: Defining Asymptotes and Infinity 3:29
- How to Solve the Difference Quotient
- How to Find the Difference Quotient with Fractions
- Go to Limits in AP Calculus: Help and Review

- Pennsylvania Grades 4-8 - English Language Arts Subject Concentration: Practice & Study Guide
- Praxis Reading for Virginia Educators - Elementary & Special Education: Practice & Study Guide
- Chemistry Lesson Plans & Activities
- Certified Fraud Examiner (CFE): Exam Prep & Study Guide
- PMI-PBA Certification Exam Study Guide - PMI Professional in Business Analysis
- International Marketing Basics
- PMI-PBA Certification Overview
- Business Report & Proposal Basics
- Types of Academic Essays
- Prisons in the United States
- WV Next Generation Standards for Science
- Response to Intervention (RTI) in Georgia
- The PMI-PBA Certification Process
- WIDA Can Do Descriptors for Grades 6-8
- PMI-PBA Exam: Policies, Procedures & Results
- 14 Cows for America Lesson Plan
- ELL Services in Illinois

- Implications of Business Strategy for Training
- Tragic Flaw in Macbeth: Quotes & Analysis
- Animal Farm Windmill: Symbolism & Analysis
- Supply Chain Sustainability: Environmentally Sound Choices
- Richard Feynman: Biography & Quotes
- Short Stories About Space for Kids
- Movie Review Lesson Plan for ESL Students
- Patient Care Data Set (PCDS): Purpose & Components
- Quiz & Worksheet - Women in the U.S. Revolutionary War
- Quiz & Worksheet - Lattice Energy
- Quiz & Worksheet - Lexically Ambiguous Sentences
- Quiz & Worksheet - What are Achievement Tests?
- Quiz & Worksheet - Cost Accounting & Corporate Mission
- Number Sense for Business Flashcards
- Math for Financial Analysis Flashcards

- Measuring Customer Satisfaction
- Important People in World History Study Guide
- NES Essential Academic Skills - Reading: Test Practice & Study Guide
- NMTA Elementary Education Subtest I: Practice & Study Guide
- Philosophy 103: Ethics - Theory & Practice
- Quadratic Functions & Polynomials
- Influences on Climate
- Quiz & Worksheet - Practice Asking ~'Where Am I?~' in Spanish
- Quiz & Worksheet - Stereotypes & Types of Information Processing
- Quiz & Worksheet - Glomerulus
- Quiz & Worksheet - The US Home Front During WWII
- Quiz & Worksheet - Events in the US in 1968

- Spanish Grammar: Imperfect vs. Preterite for Events and Actions
- Dome of the Rock: Definition, History, Architecture & Facts
- Summer Tutoring Ideas
- NYSTCE Score Report Information
- Average LSAT Score
- Summer Tutoring Ideas
- Finding Summer Teaching Opportunities
- 13 Colonies Lesson Plan
- Curriculum Vitae Template
- Study.com Demo for Enterprise
- Alliteration Lesson Plan
- GACE Test Dates & Registration

Browse by subject