Call Now to Set Up Tutoring:

(888) 888-0446

Calculus AB : Calculus AB

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 #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

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

View Tutors

Mahan
Certified Tutor

The University of Texas at Austin, Bachelor of Science, Chemical Engineering. The University of Texas at Austin, Bachelor of ...

View Tutors

Ethan
Certified Tutor

University of Richmond, Bachelor in Arts, Physics.

View Tutors

Rebecca
Certified Tutor

University of North Carolina at Greensboro, Bachelor of Science, Human Development and Family Studies. Northcentral Universit...

All Calculus AB Resources

45 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Statistics Tutors in Chicago, Computer Science Tutors in Denver, Reading Tutors in Washington DC, SAT Tutors in Phoenix, SSAT Tutors in Seattle, Physics Tutors in Boston, GMAT Tutors in Seattle, Reading Tutors in New York City, LSAT Tutors in San Diego, Reading Tutors in Seattle

Popular Courses & Classes

MCAT Courses & Classes in Boston, GRE Courses & Classes in San Diego, GRE Courses & Classes in Boston, GMAT Courses & Classes in Atlanta, GRE Courses & Classes in Atlanta, MCAT Courses & Classes in San Diego, MCAT Courses & Classes in Denver, GRE Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in Washington DC, GMAT Courses & Classes in Miami

Popular Test Prep

LSAT Test Prep in Denver, ISEE Test Prep in Chicago, ACT Test Prep in New York City, ACT Test Prep in Miami, GMAT Test Prep in San Diego, GMAT Test Prep in Chicago, ACT Test Prep in San Francisco-Bay Area, SAT Test Prep in Phoenix, LSAT Test Prep in Houston, MCAT Test Prep in Seattle

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 : Calculus AB

Study concepts, example questions & explanations for Calculus AB

All Calculus AB Resources

Example Questions

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

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

All Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: