Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Partial Derivatives

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 #1 : Differentials

Find the total differential , , of the function

z=e^x + cos (y)

Possible Answers:

dz=xe^x dx+sin(y)dy

dz=e^x dx-sin(y)dy

dz=cos(y) dx-e^xdy

dz=e^x dx+sin(y)dy

Correct answer:

dz=e^x dx-sin(y)dy

Explanation:

The total differential is defined as

$dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$

We first find

$\frac{\partial z}{\partial x}$

by taking the derivative with respect to and treating as a constant.

$\frac{\partial z}{\partial x}=e^x$

We then find

$\frac{\partial z}{\partial y}$

by taking the derivative with respect to and treating as a constant.

$\frac{\partial z}{\partial y}=-sin(y)$

We then substitute these partial derivatives into the first equation to get the total differential

dz=e^x dx-sin(y)dy

Report an Error

Example Question #3 : Differentials

Find the total differential, , of the function

$z=e^{x^2+y^2}+sec(2x)$

Possible Answers:

$dz=[2sec(2x)tan(2x)]dx+2ye^{x^2+y^2}dy$

$dz=[2xe^{x^2+y^2}]dx+2ye^{x^2+y^2}dy$

$dz=[2xe^{x^2+y^2}+2sec(2x)tan(2x)]dx+2ye^{x^2+y^2}dy$

Correct answer:

$dz=[2xe^{x^2+y^2}+2sec(2x)tan(2x)]dx+2ye^{x^2+y^2}dy$

Explanation:

The total differential is defined as

$dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$

We first find $\frac{\partial z}{\partial x}$ by taking the derivative with respect to and treating as a constant.

$\frac{\partial z}{\partial x}=\frac{\partial }{\partial x}[e^{x^2+y^2}+sec(2x)]=2xe^{x^2+y^2}+2sec(2x)tan(2x)$

We then find $\frac{\partial z}{\partial y}$ by taking the derivative with respect to and treating as a constant.

$\frac{\partial z}{\partial y}=\frac{\partial }{\partial y}[e^{x^2+y^2}+sec(2x)]=2ye^{x^2+y^2}$

We then substitute these partial derivatives into the first equation to get the total differential

$dz=[2xe^{x^2+y^2}+2sec(2x)tan(2x)]dx+2ye^{x^2+y^2}dy$

Report an Error

Example Question #4 : Differentials

Find the total differential, , of the function

$w=e^{x+y}tan(4z)+sin(z)$

Possible Answers:

$dw=e^{x+y}tan(4z)dx+e^{x+y}tan(4z)}dy +[4e^{x+y}sec^2(4z)+cos(z)]dz$

$dw=e^{x+y}tan(4z)dx+e^{x+y}tan(4z)}dy +[sec^2(4z)]dz$

$dw=e^{x+y}dx+e^{x+y}}dy +[4e^{x+y}sec^2(4z)+cos(z)]dz$

Correct answer:

$dw=e^{x+y}tan(4z)dx+e^{x+y}tan(4z)}dy +[4e^{x+y}sec^2(4z)+cos(z)]dz$

Explanation:

The total differential is defined as

$dw=\frac{\partial w}{\partial x}dx+\frac{\partial w}{\partial y}dy+\frac{\partial w}{\partial z}dz$

We first find $\frac{\partial w}{\partial x}$ by taking the derivative with respect to and treating the other variables as constants.

$\frac{\partial w}{\partial x}=\frac{\partial }{\partial x}[e^{x+y}tan(4z)+sin(z)]=e^{x+y}tan(4z)$

We then find $\frac{\partial w}{\partial y}$ by taking the derivative with respect to and treating the other variables as constants.

$\frac{\partial w}{\partial y}=\frac{\partial }{\partial y}[e^{x+y}tan(4z)+sin(z)]=e^{x+y}tan(4z)$

We then find $\frac{\partial w}{\partial z}$ by taking the derivative with respect to and treating the other variables as constants.

$\frac{\partial w}{\partial z}=\frac{\partial }{\partial y}[e^{x+y}tan(4z)+sin(z)]=4e^{x+y}sec^2(4z)+cos(z)$

We then substitute these partial derivatives into the first equation to get the total differential

$dw=e^{x+y}tan(4z)dx+e^{x+y}tan(4z)}dy +[4e^{x+y}sec^2(4z)+cos(z)]dz$

Report an Error

Example Question #1 : Differentials

Find the total derivative of the function:

$f(x, y)=x\sec(y)+6y^4$

Possible Answers:

$(\sec(y)+x\sec(y)\tan(y)+24y^3)dy$

$\sec(y)dx+(x\sec(y)\tan(y)+24y^3)dy$

$\sec(y)dy+(x\sec(y)\tan(y)+24y^3)dx$

$-\sec(y)dx+(x\sec(y)\tan(y)+24y^3)dy$

Correct answer:

$\sec(y)dx+(x\sec(y)\tan(y)+24y^3)dy$

Explanation:

The total derivative of a function of two variables is given by the following:

f_x dx+f_ydy

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives of the function are

$f_x=\sec(y)$

$f_y=x\sec(y)\tan(y)+24y^3$

The derivatives were found using the following rules:

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

So, our final answer is

$\sec(y)dx+(x\sec(y)\tan(y)+24y^3)dy$

Report an Error

Example Question #1 : Differentials

Find the total derivative of the function:

$f(x, y)=\sqrt{x+y^2}+\ln(x^2)$

Possible Answers:

$(\frac{1}{2\sqrt{x+y^2}}+\frac{1}{x})dx+(\frac{y}{\sqrt{x+y^2}})dy$

$(\frac{1}{2\sqrt{x+y^2}}+\frac{2}{x})dy+(\frac{y}{\sqrt{x+y^2}})dx$

$(\frac{1}{2\sqrt{x+y^2}}+\frac{2}{x})dx+(\frac{y}{\sqrt{x+y^2}})dy$

$\frac{1}{2\sqrt{x+y^2}}+\frac{2}{x}+\frac{y}{\sqrt{x+y^2}}$

Correct answer:

$(\frac{1}{2\sqrt{x+y^2}}+\frac{2}{x})dx+(\frac{y}{\sqrt{x+y^2}})dy$

Explanation:

The total derivative of a function of two variables is given by

f_xdx+f_ydy

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

$f_x=\frac{1}{2\sqrt{x+y^2}}+\frac{2}{x}$

$f_y=\frac{y}{\sqrt{x+y^2}}$

The derivatives were found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \ln(u)=\frac{1}{u}\frac{\mathrm{d} u}{\mathrm{d} x}$

Report an Error

Example Question #7 : Differentials

Find the total derivative of the function:

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

Possible Answers:

$-y^2\sin(xy^2)dx-2xy\sin(xy^2)dy+2ze^{z^2}dz$

$-y^2\sin(xy^2)-2xy\sin(xy^2)+2ze^{z^2}$

$-y^2\sin(xy^2)dx-2xy\sin(xy^2)dy$

$-y^2\sin(xy^2)dx+2xy\sin(xy^2)dy+2ze^{z^2}dz$

Correct answer:

$-y^2\sin(xy^2)dx-2xy\sin(xy^2)dy+2ze^{z^2}dz$

Explanation:

The total derivative of a function f(x, y, z) is given by

f_xdx+f_ydy+f_zdz

So, we must find the partial derivatives. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partials are

$f_x=-y^2\sin(xy^2)$

$f_y=-2xy\sin(xy^2)$

$f_z=2ze^{z^2}$

The derivatives were found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$ $\frac{\mathrm{d} }{\mathrm{d} x} ax=a$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u\frac{\mathrm{d} u}{\mathrm{d} x}$

Report an Error

Example Question #1 : Differentials

Find the total derivative of the following function:

$f(x, y,z)=x^2y+\csc(z)$

Possible Answers:

$2xydx+x^2dy+\csc(z)\cot(z)dz$

$2xy+x^2-\csc(z)\cot(z)$

$2xydx+x^2dy-\csc(z)\cot(z)dz$

$2xdx-\csc(z)\cot(z)dz$

Correct answer:

$2xydx+x^2dy-\csc(z)\cot(z)dz$

Explanation:

The total derivative of a function is given by

f_xdx+f_ydy+f_zdz

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

f_x=2xy

f_y=x^2

$f_z=-\csc(z)\cot(z)$

The derivatives were found using the following rules:

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

Report an Error

Example Question #9 : Differentials

Find the differential of the following function:

$f(x, y, z)=x^5+\tan(y^2z)$

Possible Answers:

$5x^4dx+2yz\sec^2(y^2z)dy+y^2\sec^2(y^2z)dz$

$5x^4dx+y\sec^2(y^2z)dy+z\sec^2(y^2z)dz$

$5x^4dx+2yz\sec^2(y^2z)dy+zy^2\sec^2(y^2z)dz$

$5x^4+\sec^2(y^2z)(2yz+y^2)$

Correct answer:

$5x^4dx+2yz\sec^2(y^2z)dy+y^2\sec^2(y^2z)dz$

Explanation:

The differential of a function is given by

f_xdx+f_ydy+f_zdz

So, we must find the partial derivatives of the function. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

f_x=5x^4

$f_y=2yz\sec^2(y^2z)$

$f_z=y^2\sec^2(y^2z)$

The derivatives were found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \tan(x)=\sec^2(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} ax=a$

Report an Error

Example Question #10 : Differentials

Find the differential of the function

$f(x, y, z)=ze^y+\ln(x^2)$

Possible Answers:

$\frac{2}{x}+e^y(z+1)$

$\frac{2}{x}dx+ze^ydy+e^ydz$

$\frac{2}{x}dx+e^ydy+ze^ydz$

$\frac{2}{x^2}dx+ze^ydy+e^ydz$

Correct answer:

$\frac{2}{x}dx+ze^ydy+e^ydz$

Explanation:

The differential of a function is given by

f_xdx+f_ydy+f_zdz

The partial derivatives of the function are

$f_x=\frac{2}{x}$

f_y=ze^y

f_z=e^y

The derivatives were found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} \ln(u)=\frac{1}{u}\frac{\mathrm{d} u}{\mathrm{d} x}$ , $\frac{\mathrm{d} }{\mathrm{d} x}x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^x=e^x$ , $\frac{\mathrm{d} }{\mathrm{d} x} ax=a$

Report an Error

Example Question #11 : Differentials

Find the differential of the function:

g=yz+xe^x

Possible Answers:

(e^x+xe^x)dx+zdy+ydz

2e^xdx+zdy+ydz

(e^x+xe^x)dx+ydy+zdz

(e^x+xe^x)dx+zdy

Correct answer:

(e^x+xe^x)dx+zdy+ydz

Explanation:

The differential of the function is given by

f_xdx+f_ydy+f_zdz

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

f_x=e^x+xe^x

f_y=z

f_z=y

Report an Error

View Calculus 3 Tutors

Michael
Certified Tutor

University of California-San Diego, Bachelors, Physics with specialization in Earth Science. University of California-San Die...

View Calculus 3 Tutors

Gregory
Certified Tutor

Pennsylvania State University-Main Campus, Bachelor of Science, Chemical Engineering. Carnegie Mellon University, Doctor of P...

View Calculus 3 Tutors

Muhammad
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Mechanical Engineering.

All Calculus 3 Resources

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

Popular Subjects

Algebra Tutors in Atlanta, French Tutors in Chicago, French Tutors in Seattle, Reading Tutors in Washington DC, Spanish Tutors in Washington DC, Reading Tutors in Atlanta, Biology Tutors in Atlanta, English Tutors in Los Angeles, Spanish Tutors in Dallas Fort Worth, GRE Tutors in Miami

Popular Courses & Classes

LSAT Courses & Classes in Miami, ACT Courses & Classes in Washington DC, MCAT Courses & Classes in Denver, SAT Courses & Classes in Washington DC, ISEE Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in Atlanta, GMAT Courses & Classes in San Diego, Spanish Courses & Classes in Philadelphia, SSAT Courses & Classes in Los Angeles, Spanish Courses & Classes in Atlanta

Popular Test Prep

LSAT Test Prep in Denver, LSAT Test Prep in Houston, SAT Test Prep in Denver, MCAT Test Prep in Philadelphia, LSAT Test Prep in Phoenix, GRE Test Prep in Chicago, MCAT Test Prep in Miami, GMAT Test Prep in San Francisco-Bay Area, LSAT Test Prep in Washington DC, ACT Test Prep in Miami

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 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #1 : Differentials

Example Question #3 : Differentials

Example Question #4 : Differentials

Example Question #1 : Differentials

Example Question #1 : Differentials

Example Question #7 : Differentials

Example Question #1 : Differentials

Example Question #9 : Differentials

Example Question #10 : Differentials

Example Question #11 : Differentials

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: