Call Now to Set Up Tutoring:

(888) 888-0446

Calculus AB : Apply the Product Rule and Quotient Rule

Study concepts, example questions & explanations for Calculus AB

Create An Account

All Calculus AB Resources

45 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Apply The Product Rule And Quotient Rule

Which of the following is the Product Rule?

Possible Answers:

$\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - g'(x)f(x)}{g(x)}$

$\frac{d}{dx} [f(x)g(x)]= f'(x)g(x) + g'(x)f(x)$

$\frac{d}{dx} [f(x)g(x)]= f'(x)g'(x) + g(x)f(x)$

$\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - g'(x)f(x)}{g(x)^2}$

Correct answer:

$\frac{d}{dx} [f(x)g(x)]= f'(x)g(x) + g'(x)f(x)$

Explanation:

When two different functions ( f(x) and g(x) ) are being multiplied together and we need to use what we call the product rule to find the derivative. For the product rule you take the derivative of one function at a time while keeping the other constant and sum these together. The formula is: $\frac{d}{dx} [f(x)g(x)]= f'(x)g(x) + g'(x)f(x)$

Report an Error

Example Question #2 : Apply The Product Rule And Quotient Rule

If we want to find the derivative of the function f(x) = (x^2 - 2)(4x+3) , which of the following would we use?

Possible Answers:

Chain Rule

None of the above

Quotient Rule

Product Rule

Correct answer:

Product Rule

Explanation:

We would use the product rule in this case. Let g(x) = x^2 - 2 and h(x) = 4x + 3 .

We will use the formula $\frac{d}{dx}[g(x)h(x) = g'(x)h(x) + g(x)h'(x)$

$\frac{d}{dx}[g(x)h(x)] = g'(x)h(x) + g(x)h'(x)$

$\frac{d}{dx}[(x^2 -2)(4x+3)] = 2x(4x+3) + 4(x^2-2)$

$\frac{d}{dx}[(x^2 -2)(4x+3)] = 8x^2 + 6x + 4x^2 - 8$

$\frac{d}{dx}[(x^2 -2)(4x+3)] = 12x^2 + 6x - 8$

Report an Error

Example Question #3 : Apply The Product Rule And Quotient Rule

Find the derivative (do not solve) of the following function, f(x) = 4xsin(x) .

Possible Answers:

$\frac{d}{dx}[4xsin(x)] = 4[xcos(x) + sin(x)]$

$\frac{d}{dx}[4xsin(x)] = 4[xsin(x) + sin(x)]$

$\frac{d}{dx}[4xsin(x)] = 4[xcos(x) + cos(x)]$

$\frac{d}{dx}[4xsin(x)] = 4[xsin(x) + cos(x)]$

Correct answer:

$\frac{d}{dx}[4xsin(x)] = 4[xcos(x) + sin(x)]$

Explanation:

We will let g(x) = 4x and h(x) = sin(x) .

$\frac{d}{dx}[g(x)h(x)] = g'(x)h(x) + g(x)h'(x)$

$\frac{d}{dx}[4xsin(x)] = 4sin(x) + 4xcos(x)$

$\frac{d}{dx}[4xsin(x)] = 4[xcos(x) + sin(x)]$

Report an Error

Example Question #4 : Apply The Product Rule And Quotient Rule

Find the derivative (do not solve) of the following function, f(x) =(5x^2 + 8x)(7x - 2) .

Possible Answers:

$\frac{d}{dx}[(5x^2+8x)(7x-2)] = 105x^2 + 112x - 36$

$\frac{d}{dx}[(5x^2+8x)(7x-2)] = 105x^2 + 112x + 36$

$\frac{d}{dx}[(5x^2+8x)(7x-2)] = 56x^2 + 34x - 16$

$\frac{d}{dx}[(5x^2+8x)(7x-2)] = 112x^2 + 105x - 36$

Correct answer:

$\frac{d}{dx}[(5x^2+8x)(7x-2)] = 105x^2 + 112x - 36$

Explanation:

We will let g(x) = 5x^2 + 8x and h(x) = 7x - 2 .

$\frac{d}{dx}[g(x)h(x)] = g'(x)h(x) + g(x)h'(x)$

$\frac{d}{dx}[(5x^2+8x)(7x-2)] = (10x+8)(7x-2) + (5x^2+8x)(7)$

$\frac{d}{dx}[(5x^2+8x)(7x-2)] = 70x^2 - 20 + 56x - 16 + 35x^2 + 56x$

$\frac{d}{dx}[(5x^2+8x)(7x-2)] = 105x^2 + 112x - 36$

Report an Error

Example Question #1 : Apply The Product Rule And Quotient Rule

Which of the following would we use to find the derivative of the function $f(x) = \frac{x^2}{x-2}$

Possible Answers:

Quotient Rule

Product Rule

None of the above

Chain Rule

Correct answer:

Quotient Rule

Explanation:

When trying to find the derivative of the quotient of two functions, we use what is called the quotient rule. The quotient rule is similar to the product rule except you are taking the difference of two products and then dividing them further.

Report an Error

Example Question #1 : Apply The Product Rule And Quotient Rule

Which of the following is the correct formula for the quotient rule?

Possible Answers:

$\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$

$\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{f'(x)g'(x) - f(x)g(x)}{g(x)}$

$\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)}$

$\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{2g(x)}$

Correct answer:

$\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$

Explanation:

If two functions are differentiable, then so is their quotient. So we use the quotient rule to find the derivative of the entire function.

Report an Error

Example Question #6 : Apply The Product Rule And Quotient Rule

Find the derivative of the function $f(x) = \frac{e^x}{x^2}$

Possible Answers:

$\frac{d}{dx}[\frac{e^x}{x^2}] = \frac{x^3 - 2x}{e^2x}$

$\frac{d}{dx}[\frac{e^x}{x^2}] = \frac{xe^x - 2e^x}{x}$

$\frac{d}{dx}[\frac{e^x}{x^2}] = \frac{x^3 - 2x}{e^x}$

$\frac{d}{dx}[\frac{e^x}{x^2}] = \frac{xe^x - 2e^x}{x^3}$

Correct answer:

$\frac{d}{dx}[\frac{e^x}{x^2}] = \frac{xe^x - 2e^x}{x^3}$

Explanation:

We will need to use the quotient rule here. Let g(x) = e^x and h(x) = x^2 .

$\frac{d}{dx}[\frac{g(x)}{h(x)}] = \frac{g'(x)h(x) - g(x)h'(x)}{h(x)^2}$

$\frac{d}{dx}[\frac{e^x}{x^2}] = \frac{e^x(x^2) - e^x(2x)}{(x^2)^2}$

$\frac{d}{dx} [\frac{e^x}{x^2}] = \frac{x(xe^x - 2e^x)}{x^4}$

$\frac{d}{dx}[\frac{e^x}{x^2}] = \frac{xe^x - 2e^x}{x^3}$

Report an Error

Example Question #7 : Apply The Product Rule And Quotient Rule

Find the derivative of the function $f(x) = \frac{2x^3 - 9x}{x^4}$ .

Possible Answers:

$\frac{d}{dx}[\frac{2x^3 - 9x}{x^4}] = \frac{28x^3 + 6x^2 - 9}{x^4}$

$\frac{d}{dx}[\frac{2x^3 - 9x}{x^4}] = \frac{7x^3 + x^2 - 3}{x^4}$

$\frac{d}{dx}[\frac{2x^3 - 9x}{x^4}] = \frac{28x^3 + 6x^2 - 9}{x^8}$

$\frac{d}{dx}[\frac{2x^3 - 9x}{x^4}] = \frac{x^3 + x^2}{x^4}$

Correct answer:

$\frac{d}{dx}[\frac{2x^3 - 9x}{x^4}] = \frac{28x^3 + 6x^2 - 9}{x^4}$

Explanation:

We will use the quotient rule to find the derivative. Let g(x) = 2x^3 - 9x and h(x) = x^4 .

$\frac{d}{dx}[\frac{g(x)}{h(x)}] = \frac{g'(x)h(x) - g(x)h'(x)}{h(x)^2}$

$\frac{d}{dx}[\frac{2x^3 - 9x}{x^4}] = \frac{(6x^2 - 9)x^4 - (2x^3 - 9x)4x^3}{(x^4)^2}$

$\frac{d}{dx}[\frac{2x^3 - 9x}{x^4}] = \frac{6x^6 - 9x^4 - 8x^7 + 36x^7}{x^8}$

$\frac{d}{dx}[\frac{2x^3 - 9x}{x^4}] =\frac{28x^7 + 6x^6 - 9x^4}{x^8}$

$\frac{d}{dx}[\frac{2x^3 - 9x}{x^4}] = \frac{x^4(28x^3 + 6x^2 - 9)}{x^8}$

$\frac{d}{dx}[\frac{2x^3 - 9x}{x^4}] = \frac{28x^3 + 6x^2 - 9}{x^4}$

Report an Error

Example Question #8 : Apply The Product Rule And Quotient Rule

True or False: You may have to use the quotient and power rules for the same function.

Possible Answers:

True

False

Correct answer:

True

Explanation:

Take the function $f(x) = \frac{xe^x}{3 - x}$ for example. We would have to use the product rule at first for the numerator and once that is simplified we would need to use the quotient rule to find the derivative of the entire function.

Report an Error

Example Question #1 : Apply The Product Rule And Quotient Rule

True or False: For a function in the form of f(x) = g(x)h(x) , if a derivative does not exist for one of g(x) or h(x) but exists for the other, then you are still able to use the product rule to find the derivative of the entire function.

Possible Answers:

False

True

Correct answer:

False

Explanation:

This is not true. In order to use the product rule you need to be able to derive each of the functions and then sum them. So if the derivative does not exist for one of these functions then you would not be able to use the product rule.

Report an Error

View Tutors

Ronald
Certified Tutor

Towson University, Master of Science, Counseling Psychology.

View Tutors

Jonathan
Certified Tutor

East Carolina University, Bachelor of Science, Chemistry. East Carolina University, Bachelor of Science, Biology, General.

View Tutors

Constance
Certified Tutor

Loyola University-Chicago, Bachelors, Psychology. Sam Houston State University, Masters, Counseling Psychology.

All Calculus AB Resources

45 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

LSAT Tutors in Atlanta, Algebra Tutors in New York City, Reading Tutors in Los Angeles, SAT Tutors in Miami, Chemistry Tutors in Dallas Fort Worth, Calculus Tutors in Los Angeles, French Tutors in New York City, Math Tutors in San Francisco-Bay Area, Spanish Tutors in Dallas Fort Worth, Computer Science Tutors in Philadelphia

Popular Courses & Classes

Spanish Courses & Classes in San Diego, GMAT Courses & Classes in Houston, LSAT Courses & Classes in Chicago, SAT Courses & Classes in New York City, SAT Courses & Classes in Washington DC, GMAT Courses & Classes in Phoenix, MCAT Courses & Classes in San Francisco-Bay Area, GMAT Courses & Classes in Denver, GMAT Courses & Classes in New York City, GMAT Courses & Classes in Philadelphia

Popular Test Prep

ISEE Test Prep in San Diego, SSAT Test Prep in Boston, SAT Test Prep in San Diego, LSAT Test Prep in Chicago, LSAT Test Prep in Houston, GMAT Test Prep in Philadelphia, GRE Test Prep in Boston, SSAT Test Prep in New York City, SAT Test Prep in Washington DC, ISEE Test Prep in Chicago

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Calculus AB : Apply the Product Rule and Quotient Rule

Study concepts, example questions & explanations for Calculus AB

All Calculus AB Resources

Example Questions

Example Question #1 : Apply The Product Rule And Quotient Rule

Example Question #2 : Apply The Product Rule And Quotient Rule

Example Question #3 : Apply The Product Rule And Quotient Rule

Example Question #4 : Apply The Product Rule And Quotient Rule

Example Question #1 : Apply The Product Rule And Quotient Rule

Example Question #1 : Apply The Product Rule And Quotient Rule

Example Question #6 : Apply The Product Rule And Quotient Rule

Example Question #7 : Apply The Product Rule And Quotient Rule

Example Question #8 : Apply The Product Rule And Quotient Rule

Example Question #1 : Apply The Product Rule And Quotient Rule

All Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: