We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

Create An Account

All Calculus 3 Resources

6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #973 : Partial Derivatives

Find $f_{xzy}$ of the function x^3yz+z^5+xy^5

Possible Answers:

xy^2+5x

3x^2z

3x^2

Correct answer:

3x^2

Explanation:

To find $f_{xzy}$ , you must take $\frac{\partial }{\partial x},\frac{\partial }{\partial z},\frac{\partial }{\partial y}$ consecutively of the function. In doing so, we get $\frac{\partial }{\partial x}(x^3yz+z^5+xy^5)=3x^2yz+y^5$ , $\frac{\partial }{\partial z}(3x^2yz+y^5)=3x^2y$ , and finally $\frac{\partial }{\partial y}(3x^2y)=3x^2$

Report an Error

Example Question #974 : Partial Derivatives

Find $f_{xzx}$ of the following function:

$f(x, z)=x^2\cos(z^2)$

Possible Answers:

$-4xz\sin(z^2)$

$4z\sin(z^2)$

$-4z\sin(z^2)$

$-2z\sin(z^2)$

Correct answer:

$-4z\sin(z^2)$

Explanation:

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

To start, we must find the partial derivative of the function with respect to x:

$f_x=2x\cos(z^2)$

The derivative was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} ax=a$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Next, we find the partial derivative of the above function with respect to z:

$f_{xz}=-4xz\sin(z^2)$

The derivative was found using rules above as well as the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$

Finally, we take the partial derivative of the above function with respect to x:

$f_{xzx}=-4z\sin(z^2)$

The rule used to find the partial derivative is above.

Report an Error

Example Question #975 : Partial Derivatives

Find $f_{xxy}$ of the following function:

$f(x,y,z)= (\cos x)y^3e^\tan z$

Possible Answers:

$f_{xxy}=-3y^2e^ {tan z}(\cos x)(\sec^2 z)$

$f_{xxy}=y^2e^ {tan z}(\sin x)$

$f_{xxy}=-3y^2e^ {tan z}(\cos x)$

$f_{xxy}=-3y(\cos x)$

Correct answer:

$f_{xxy}=-3y^2e^ {tan z}(\cos x)$

Explanation:

The first thing to do is to take one partial derivative with respect to only one variable while everything else behaves as a constant. In this case, you can start by taking the partial derivative of one , which would be:

$f_{x}=-y^3e^{\tan z}(\sin x)$

The derivative was found by using the following rule:

$\frac{d}{dx}\cos x=-\sin x$

Then, you can take the partial derivative with respect to again because is expressed twice. This partial derivative is:

$f_{xx}=-y^3e^{\tan z}(\cos x)$

The derivative was found by using the following rule:

$\frac{d}{dx}\sin x=\cos x$

Finally, you must take the partial derivative with respect to which is:

$f_{xxy}=-3y^2e^{\tan z}(\cos x)$

The derivative was taken by using the following rule:

$\frac{d}{dx}x^n=nx^{n-1}$

Report an Error

Example Question #976 : Partial Derivatives

Find $f_{xyz}$ of the following function:

f(x,y,z)=5x^4y^3e^z

Possible Answers:

$f_{xyz}=20x^3y^2e^z$

$f_{xyz}=60x^3y^2e^z$

$f_{xyz}=360x^3ye^z$

$f_{xyz}=60x^3y^2ze^z$

Correct answer:

$f_{xyz}=60x^3y^2e^z$

Explanation:

$f_{x}=20x^3y^3e^z$

The derivative was found by using the following rule:

$\frac{d}{dx}x^n=nx^{n-1}$

Then, you can take the partial derivative with respect to . This partial derivative is:

$f_{xy}=60x^3y^2e^z$

The derivative was found by using the following rule:

$\frac{d}{dx}x^n=nx^{n-1}$

Finally, you must take the partial derivative with respect to which is:

$f_{xyz}=60x^3y^2e^z$

The derivative was taken by using the following rule:

$\frac{d}{dx}e^x=e^x\cdot x'$

Report an Error

Example Question #977 : Partial Derivatives

Find $f_{yyz}$ of the following function:

$f(x,y,z)=\cos xe^{2y}\sqrt{z}$

Possible Answers:

$\frac{-2\sin xe^{2y}}{\sqrt{z}}$

$\frac{4\cos xe^{2y}}{\sqrt{z}}$

${2\cos xe^{2y}}{\sqrt{z}}$

$\frac{2\cos xe^{2y}}{\sqrt{z}}$

Correct answer:

$\frac{2\cos xe^{2y}}{\sqrt{z}}$

Explanation:

$f_{y}=2\cos xe^{2y}\sqrt{z}$

The derivative was found by using the following rule:

$\frac{d}{dx} e^x=e^x\cdot x'$

Then, you can take the partial derivative with respect to again because is expressed twice. This partial derivative is:

$f_{yy}=4\cos xe^{2y}\sqrt{z}$

The derivative was found by using the following rule:

$\frac{d}{dx}e^x=e^x\cdot x'$

Finally, you must take the partial derivative with respect to which is:

$f_{yyz}=4\cos xe^{2y}(\frac{1}{2}z^{\frac{-1}{2}})=\frac{2\cos xe^{2y}}{\sqrt{z}}$

The derivative was taken by using the following rule:

$\frac{d}{dx}x^n=nx^{n-1}$

Report an Error

Example Question #978 : Partial Derivatives

Find $f_{xzy}$ of the following function:

$f(x,y,z)=y^2e^{tan z}\cos x$

Possible Answers:

$f_{xzy}=-2ye^{\tan z}\sec^2 z\sin x$

$f_{xzy}=-2ye^{\tan z} \sin x$

$f_{xzy}=-2y^2e^{\tan z}\sec^2 z\sin x$

$f_{xzy}=-2ye^{\tan z}\sec^2 z\cos x$

Correct answer:

$f_{xzy}=-2ye^{\tan z}\sec^2 z\sin x$

Explanation:

The first thing to do is to take the partial derivative with respect to one variable while everything else behaves as a constant. In this case, you can take the partial derivative with respect to first which would be:

$f_{x}=-y^2e^{tan z}\sin x$

Then, you can take the partial derivative of which would be:

$f_{xz}=-y^2e^{tan z}\sec^2 z\sin x$

Finally, you must take the partial derivative with respect to :

$f_{xzy}=-2ye^{tan z}\sec^2 z\sin x$

Report an Error

Example Question #979 : Partial Derivatives

Find $f_{xxyz}$ of the following function:

$f(x,y,z)=5x^3y^4\sqrt{z}$

Possible Answers:

${60xy^4}{\sqrt{z}}$

$\frac{60xy^3}{\sqrt{z}}$

$\frac{120xy^3}{\sqrt{z}}$

${60xy^3}{\sqrt{z}}$

Correct answer:

$\frac{60xy^3}{\sqrt{z}}$

Explanation:

The first thing to do is to find the partial derivative with respect to one variable while everything else behaves as a constant. In this case, you can start by taking the partial derivative with respect to which is:

$15x^2y^4\sqrt{z}$

Then, you can take the partial derivative with respect to again because is expressed twice in the problem:

$30xy^4\sqrt{z}$

Then, you can take the partial derivative with respect to which is:

$120xy^3\sqrt{z}$

Finally, you must take the partial derivative with respect to which is:

$120xy^3(\frac{1}{2}z^\frac{-1}{2})=\frac{60xy^3}{\sqrt{z}}$

Report an Error

Example Question #980 : Partial Derivatives

Find $f_{yyz}$ of the following function:

$f(x,y,z)=z^3x\ln y$

Possible Answers:

$f_{yyz}=\frac{3z^2x}{y^2}$

$f_{yyz}=\frac{-3z^2x}{y^2}$

$f_{yyz}=\frac{-3z^2x}{y}$

$f_{yyz}=\frac{-3zx}{y^2}$

Correct answer:

$f_{yyz}=\frac{-3z^2x}{y^2}$

Explanation:

The first thing to do is to take the partial derivative of one variable while everything else behaves as a constant. In this case, you can take the partial derivative with respect to which is:

$f_y=\frac{z^3x}{y}$

Then, you can take the partial derivative with respect to again because the problem expresses twice. This would be:

$f_{yy}=\frac{-z^3x}{y^2}$

Finally, you must take the partial derivative with respect to which would be:

$f_{yyz}=\frac{-3z^2x}{y^2}$

Report an Error

Example Question #981 : Partial Derivatives

Find $f_{yzz}$ of the following function:

$f(x,y,z)=xy\ln z$

Possible Answers:

$\frac{-xy}{z^2}$

$\frac{x}{z}$

$\frac{-x}{z^2}$

$\frac{x}{z^2}$

Correct answer:

$\frac{-x}{z^2}$

Explanation:

The first thing to do is to find the partial derivative with respect to one variable while everything else behaves as a constant. In this case, you can take the partial derivative with respect to which is:

$f_y=x\ln z$

Then, you can take the partial derivative with respect to one which is:

$f_{yz}=\frac{x}{z}$

Finally, you must take the partial derivative with respect to again because the problem expresses it twice. This would be:

$f_{yzz}=\frac{-x}{z^2}$

Report an Error

Example Question #982 : Partial Derivatives

Find $f_{xxz}$ of the following function:

$f(x,y,z)=5yz^2\cos x$

Possible Answers:

$-10yz^2\cos x$

$-10yz\cos x$

$10yz\cos x$

$-5yz\cos x$

Correct answer:

$-10yz\cos x$

Explanation:

The first thing that you need to do is to find the partial derivative with respect to one variable while everything else behaves as a constant. In this case, you can take the partial derivative with respect to which is:

$f_x=-5yz^2\sin x$

Then, you can take the partial derivative with respect to again because it is expressed twice in the problem. This would be:

$f_{xx}=-5yz^2\cos x$

Finally, you must take the partial derivative with respect to which is:

$f_{xxz}=-10yz\cos x$

Report an Error

View Calculus 3 Tutors

Magdy
Certified Tutor

Cairo University, Egypt, Bachelor of Science, Electrical Engineering. New Mexico State University-Main Campus, Doctor of Phil...

View Calculus 3 Tutors

Corrina
Certified Tutor

Massachusetts Institute of Technology, Bachelor of Science, Mechanical Engineering.

View Calculus 3 Tutors

Ranjeeta
Certified Tutor

University of Rajasthan, Bachelor of Science, Mathematics. University of Rajasthan, Master of Science, Mathematics.

All Calculus 3 Resources

6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

GRE Tutors in Boston, SAT Tutors in Phoenix, Biology Tutors in Phoenix, Math Tutors in San Francisco-Bay Area, Biology Tutors in San Francisco-Bay Area, ACT Tutors in Atlanta, Statistics Tutors in Chicago, Statistics Tutors in Seattle, Biology Tutors in San Diego, Biology Tutors in Dallas Fort Worth

Popular Courses & Classes

GMAT Courses & Classes in Los Angeles, ACT Courses & Classes in Seattle, Spanish Courses & Classes in Philadelphia, SSAT Courses & Classes in Atlanta, GMAT Courses & Classes in San Diego, GRE Courses & Classes in San Diego, SAT Courses & Classes in Philadelphia, SAT Courses & Classes in Los Angeles, SSAT Courses & Classes in Phoenix, ACT Courses & Classes in Phoenix

Popular Test Prep

GMAT Test Prep in Washington DC, GRE Test Prep in Atlanta, ISEE Test Prep in Washington DC, ISEE Test Prep in Los Angeles, ACT Test Prep in Atlanta, SAT Test Prep in San Diego, SSAT Test Prep in New York City, ACT Test Prep in Phoenix, LSAT Test Prep in Chicago, ACT 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 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #973 : Partial Derivatives

Example Question #974 : Partial Derivatives

Example Question #975 : Partial Derivatives

Example Question #976 : Partial Derivatives

Example Question #977 : Partial Derivatives

Example Question #978 : Partial Derivatives

Example Question #979 : Partial Derivatives

Example Question #980 : Partial Derivatives

Example Question #981 : Partial Derivatives

Example Question #982 : Partial Derivatives

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: