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 #453 : Partial Derivatives

Find $\frac{\partial f}{\partial y}$ for $f \left(x,y \right )=\sin \left(x^2y-8y^2x \right )$ .

Possible Answers:

$\cos \left(x^2y-8y^2x \right ) \left(x^2-8x \right )$

$\cos \left(x^2y-8y^2x \right ) \left(x^2-16x \right )$

$\cos \left(x^2y-8y^2x \right ) \left(x^2-16yx \right )$

$\sin \left(x^2y-8y^2x \right ) \left(x^2-16yx \right )$

Correct answer:

$\cos \left(x^2y-8y^2x \right ) \left(x^2-16yx \right )$

Explanation:

When finding partial derivatives of multi-variable functions with respect to a specific variable, all other variables are treated as a constant. By the chain-rule,

$\frac{\partial }{\partial y}\sin \left(f\left(x,y \right ) \right )= \cos \left(f\left(x,y \right ) \right )\cdot f_y \left(x,y \right )$ .

Noting, that

$f \left(x,y \right )= x^2y -8y^2x \Rightarrow f_y = x^2 - 16yx$ .

Plugging all of this in, we arrive at the final result,

$\cos \left(x^2y-8y^2x \right ) \left(x^2-16yx \right )$ .

Report an Error

Example Question #454 : Partial Derivatives

Find $\frac{\partial f}{\partial x}$ of $f \left(x,y \right ) = \ln \left(\frac{x}{y} \right )$

Possible Answers:

$\frac{1}{y}$

$\frac{1}{x}$

$\frac{x}{y}$

$\frac{y}{x}$

Correct answer:

$\frac{1}{x}$

Explanation:

In finding the partial derivative of a multi-variable function with respect to a particular variable, all other variables are treated as constant. We can begin by rewriting our function in the following way

$\ln \left(\frac{x}{y} \right )= \ln x - \ln y$ .

Therefore, when we take a derivative with respect to , we find

$f_x = \frac{1}{x}$ .

Report an Error

Example Question #455 : Partial Derivatives

Find $\frac{\partial ^2 f}{\partial x \partial y}$ of $f \left(x,y \right )= 3xy^2+4x^2y$

Possible Answers:

3y^2+8xy

6y+8x

Correct answer:

6y+8x

Explanation:

When taking mixed-partials of a multi-variable function, such as $f \left(x,y \right )$ , the order in which the partials derivatives are taken does not matter. Consider the following

$\frac{\partial ^2 f}{\partial x \partial y} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y} \right )= \frac{\partial }{\partial x} \left(6xy+4x^2 \right )= 6y+8x$ .

Report an Error

Example Question #456 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{xyz}\\&\text{Where }f(x,y,z)=-\frac{ {(4x^3e^{(y^2)})}}{z^2} - 9x^3cos{(y^4)}ln{(z^4)}\end{align*}$

Possible Answers:

$\frac{{(432x^2y^3sin{(y^4)})}}{z} -\frac{ {(48x^2ye^{(y^2)})}}{z^3}$

$\frac{{(432x^2y^3sin{(y^4)})}}{z} +\frac{ {(48x^2ye^{(y^2)})}}{z^3}$

${\frac{{(8x^3e^{(y^2)})}}{z^3} -\frac{ {(12x^2e^{(y^2)})}}{z^2} -\frac{ {(36x^3cos{(y^4)})}}{z} - 27x^2cos{(y^4)}ln{(z^4)} + 36x^3y^3ln{(z^4)}sin{(y^4)} -\frac{ {(8x^3ye^{(y^2)})}}{z^2} }$

${\frac{{(432x^2y^3sin{(y^4)})}}{z} - 27x^2cos{(y^4)}ln{(z^4)} -\frac{ {(12x^2e^{(y^2)})}}{z^2} + 108x^2y^3ln{(z^4)}sin{(y^4)} -\frac{ {(24x^2ye^{(y^2)})}}{z^2} +\frac{ {(48x^2ye^{(y^2)})}}{z^3} }$

Correct answer:

$\frac{{(432x^2y^3sin{(y^4)})}}{z} +\frac{ {(48x^2ye^{(y^2)})}}{z^3}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&d[uv]=udv+vdu\\&d[cos(u)]=-sin(u)du\\&d[ln(u)]=\frac{du}{u}\\&d[e^u]=e^udu\\&f_{x}=-\frac{ {(12x^2e^{(y^2)})}}{z^2} - 27x^2cos{(y^4)}ln{(z^4)}\\&f_{xy}=108x^2y^3ln{(z^4)}sin{(y^4)} -\frac{ {(24x^2ye^{(y^2)})}}{z^2}\\&f_{xyz}=\frac{{(432x^2y^3sin{(y^4)})}}{z} +\frac{ {(48x^2ye^{(y^2)})}}{z^3}\end{align*}$

Report an Error

Example Question #457 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zx}\\&\text{Where }f(x,y,z)=9\cdot2^{(x^3)}sin{(y^3)}e^{(z)}\end{align*}$

Possible Answers:

$168.4\cdot2^{(2x^3)}x^2sin{(y^3)}^2e^{(2z)}$

${18.71\cdot2^{(3x^2)}x^2sin{(3y^2)}e^{(z)}}$

${9\cdot2^{(x^3)}sin{(y^3)}e^{(z)} + 18.71\cdot2^{(x^3)}x^2sin{(y^3)}e^{(z)}}$

${18.71\cdot2^{(x^3)}x^2sin{(y^3)}e^{(z)}}$

Correct answer:

${18.71\cdot2^{(x^3)}x^2sin{(y^3)}e^{(z)}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&d[uv]=udv+vdu\\&d[e^u]=e^udu\\&d[a^u]=a^uduln(a)\\&f_{z}=9\cdot2^{(x^3)}sin{(y^3)}e^{(z)}\\&f_{zx}=18.71\cdot2^{(x^3)}x^2sin{(y^3)}e^{(z)}\end{align*}$

Report an Error

Example Question #458 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yxz}\\&\text{Where }f(x,y,z)=-3\cdot4^{(y^4)}sin{(z^3)}ln{(x)}\end{align*}$

Possible Answers:

$\frac{-{(449.2\cdot4^{(3y^4)}y^3z^2cos{(z^3)}sin{(z^3)}^2ln{(x)}^2)}}{x}$

${\frac{-{(49.91\cdot4^{(y^4)}y^3z^2cos{(z^3)})}}{x}}$

${-\frac{ {(3\cdot4^{(y^4)}sin{(z^3)})}}{x} - 9\cdot4^{(y^4)}z^2cos{(z^3)}ln{(x)} - 16.64\cdot4^{(y^4)}y^3sin{(z^3)}ln{(x)}}$

${\frac{-{(49.91\cdot4^{(4y^3)}y^3z^2cos{(3z^2)})}}{x}}$

Correct answer:

${\frac{-{(49.91\cdot4^{(y^4)}y^3z^2cos{(z^3)})}}{x}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&d[a^u]=a^uduln(a)\\&d[ln(u)]=\frac{du}{u}\\&d[sin(u)]=cos(u)du\\&f_{y}=-16.64\cdot4^{(y^4)}y^3sin{(z^3)}ln{(x)}\\&f_{yx}=\frac{-{(16.64\cdot4^{(y^4)}y^3sin{(z^3)})}}{x}\\&f_{yxz}=\frac{-{(49.91\cdot4^{(y^4)}y^3z^2cos{(z^3)})}}{x}\end{align*}$

Report an Error

Example Question #459 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zx}\\&\text{Where }f(x,y,z)=\frac{-{(6e^{(x^3)}cos{(y)})}}{z}\end{align*}$

Possible Answers:

${\frac{{(6e^{(x^3)}cos{(y)})}}{z^2} -\frac{ {(18x^2e^{(x^3)}cos{(y)})}}{z}}$

$\frac{-{(108x^2e^{(2x^3)}cos{(y)}^2)}}{z^3}$

${\frac{{(18x^2e^{(x^3)}cos{(y)})}}{z^2}}$

${\frac{{(6e^{(x^3)}cos{(y)})}}{z^2} +\frac{ {(18x^2e^{(x^3)}cos{(y)})}}{z^2}}$

Correct answer:

${\frac{{(18x^2e^{(x^3)}cos{(y)})}}{z^2}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&d[e^u]=e^udu\\&f_{z}=\frac{{(6e^{(x^3)}cos{(y)})}}{z^2}\\&f_{zx}=\frac{{(18x^2e^{(x^3)}cos{(y)})}}{z^2}\end{align*}$

Report an Error

Example Question #460 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yzy}\\&\text{Where }f(x,y,z)=10\cdot4^{(z^2)}xsin{(y^3)}\end{align*}$

Possible Answers:

${166.4\cdot4^{(z^2)}xyzcos{(y^3)} - 249.5\cdot4^{(z^2)}xy^4zsin{(y^3)}}$

${30\cdot4^{(z^2)}xy^2cos{(y^3)} + 166.4\cdot4^{(z^2)}xyzcos{(y^3)} + 83.18\cdot4^{(z^2)}xy^2zcos{(y^3)} - 249.5\cdot4^{(z^2)}xy^4zsin{(y^3)}}$

$24955\cdot4^{(3z^2)}x^3y^4zcos{(y^3)}^2sin{(y^3)}$

${27.73\cdot4^{(z^2)}xzsin{(y^3)} + 60\cdot4^{(z^2)}xy^2cos{(y^3)}}$

Correct answer:

${166.4\cdot4^{(z^2)}xyzcos{(y^3)} - 249.5\cdot4^{(z^2)}xy^4zsin{(y^3)}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&d[sin(u)]=cos(u)du\\&d[a^u]=a^uduln(a)\\&d[cos(u)]=-sin(u)du\\&f_{y}=30\cdot4^{(z^2)}xy^2cos{(y^3)}\\&f_{yz}=83.18\cdot4^{(z^2)}xy^2zcos{(y^3)}\\&f_{yzy}=166.4\cdot4^{(z^2)}xyzcos{(y^3)} - 249.5\cdot4^{(z^2)}xy^4zsin{(y^3)}\end{align*}$

Report an Error

Example Question #461 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yxx}\\&\text{Where }f(x,y,z)=-5ln{(y^4)}e^{(z)}sin{(x)}\end{align*}$

Possible Answers:

$\frac{-{(500ln{(y^4)}^2e^{(3z)}cos{(x)}^2sin{(x)})}}{y}$

${-\frac{ {(20e^{(z)}sin{(x)})}}{y} - 10ln{(y^4)}e^{(z)}cos{(x)}}$

${\frac{{(20e^{(z)}sin{(x)})}}{y}}$

${\frac{-{(20e^{(z)}cos{(x)})}}{y}}$

Correct answer:

${\frac{{(20e^{(z)}sin{(x)})}}{y}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&d[ln(u)]=\frac{du}{u}\\&d[sin(u)]=cos(u)du\\&d[cos(u)]=-sin(u)du\\&f_{y}=\frac{-{(20e^{(z)}sin{(x)})}}{y}\\&f_{yx}=\frac{-{(20e^{(z)}cos{(x)})}}{y}\\&f_{yxx}=\frac{{(20e^{(z)}sin{(x)})}}{y}\end{align*}$

Report an Error

Example Question #2821 : Calculus 3

$\begin{align*}&\text{Perform the partial derivation, }f_{yy}\\&\text{Where }f(x,y,z)=-6cos{(z^2)}ln{(y^2)}\end{align*}$

Possible Answers:

${\frac{{(12cos{(z^2)})}}{y^2} -\frac{ {(12cos{(z^2)})}}{y}}$

${\frac{-{(24cos{(z^2)})}}{y}}$

${\frac{{(12cos{(z^2)})}}{y^2}}$

$\frac{{(144cos{(z^2)}^2)}}{y^2}$

Correct answer:

${\frac{{(12cos{(z^2)})}}{y^2}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&d[ln(u)]=\frac{du}{u}\\&f_{y}=\frac{-{(12cos{(z^2)})}}{y}\\&f_{yy}=\frac{{(12cos{(z^2)})}}{y^2}\end{align*}$

Report an Error

View Calculus 3 Tutors

Joshua
Certified Tutor

The Ohio State University, Bachelor of Science, Mathematics.

View Calculus 3 Tutors

Muhammad
Certified Tutor

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

View Calculus 3 Tutors

MrG
Certified Tutor

Long Island University-Brooklyn Campus, Master of Science, Applied Mathematics.

All Calculus 3 Resources

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

Popular Subjects

Statistics Tutors in Chicago, Statistics Tutors in Denver, SAT Tutors in Los Angeles, Reading Tutors in Philadelphia, Math Tutors in San Diego, Physics Tutors in Atlanta, Reading Tutors in San Diego, LSAT Tutors in Seattle, Algebra Tutors in Dallas Fort Worth, French Tutors in Los Angeles

Popular Courses & Classes

SSAT Courses & Classes in Seattle, ISEE Courses & Classes in Atlanta, GRE Courses & Classes in Los Angeles, ISEE Courses & Classes in Denver, ISEE Courses & Classes in Washington DC, ISEE Courses & Classes in Boston, GMAT Courses & Classes in Houston, Spanish Courses & Classes in Los Angeles, Spanish Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in Dallas Fort Worth

Popular Test Prep

GRE Test Prep in Washington DC, SAT Test Prep in Philadelphia, GMAT Test Prep in Los Angeles, LSAT Test Prep in Phoenix, LSAT Test Prep in San Francisco-Bay Area, LSAT Test Prep in San Diego, SSAT Test Prep in Miami, GRE Test Prep in Houston, SAT Test Prep in Houston, MCAT Test Prep in Houston

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 #453 : Partial Derivatives

Example Question #454 : Partial Derivatives

Example Question #455 : Partial Derivatives

Example Question #456 : Partial Derivatives

Example Question #457 : Partial Derivatives

Example Question #458 : Partial Derivatives

Example Question #459 : Partial Derivatives

Example Question #460 : Partial Derivatives

Example Question #461 : Partial Derivatives

Example Question #2821 : Calculus 3

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: