Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : How to find differential functions

Study concepts, example questions & explanations for Calculus 1

Create An Account

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1501 : Calculus

Find the slope of the function $f(x,y)=x^{-y}$ at point (3,2)

Possible Answers:

(-0.108,-0.256)

(-0.281,2.891)

(.003,1.176)

(2.234,5.781)

(-0.074,-0.122)

Correct answer:

(-0.074,-0.122)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rule will be necessary:

Derivative of an exponential: $d[a^u]=a^u\ln(a)du$

Note that may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivative of $f(x,y)=x^{-y}$ at point (3,2)

$\frac{\delta f}{\delta x}=-yx^{-y-1}=-2(3^{-2-1})=-0.074$

$\frac{\delta f}{\delta y}=-x^{-y}\ln(x)=-(3^{-2})\ln(3)=-0.122$

The slope is

(-0.074,-0.122)

Report an Error

Example Question #292 : How To Find Differential Functions

Find the slope of the function $f(x,y)=x^{\cos(y)}$ at the point $(2,\pi)$

Possible Answers:

(-0.25,0)

$(0.25,\pi)$

$(-0.25,\pi)$

$(\pi,-0.25)$

$(\pi,0.25)$

Correct answer:

(-0.25,0)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

Derivative of an exponential: $d[a^u]=a^u\ln(a)du$

Trigonometric derivative: $d[\cos(u)]=-\sin(u)du$

Note that may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivatives of $f(x,y)=x^{\cos(y)}$ at the point $(2,\pi)$

$\frac{\delta f}{\delta x}=\cos(y)x^{\cos(y)-1};\cos(\pi)(2^{\cos(\pi)-1})=-0.25$

$\frac{\delta f}{\delta y}=-\sin(y)x^{\cos(y)}\ln(x)=-\sin(\pi)2^{\cos(\pi)}\ln(2)=0$

The slope is

(-0.25,0)

Report an Error

Example Question #472 : Functions

Find the slope of the function f(x,y,z)=xz^y at point (2,3,4)

Possible Answers:

(64,96,177.4)

(64,177.4,96)

(96,64,177.4)

(96,177.4,64)

(177.4,64,96)

Correct answer:

(64,177.4,96)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rule will be necessary:

Derivative of an exponential: $d[a^u]=a^u\ln(a)du$

Note that may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivative of f(x,y,z)=xz^y at point (2,3,4)

$\frac{\delta f}{\delta x}=z^y=4^3=64$

$\frac{\delta f}{\delta y}=xz^y\ln(z)=2(4^3)\ln(4)=177.4$

$\frac{\delta f}{\delta z}=xyz^{y-1}=2(3)(4^2)=96$

The slope is

(64,177.4,96)

Report an Error

Example Question #481 : Differential Functions

Find the slope of the function f(x,y,z)=x^yy^z at point (1,2,3)

Possible Answers:

$(12,16,8\ln(2))$

$(12,8\ln(2),16)$

$(16,8\ln(2),12)$

$(16,12,8\ln(2))$

$(8\ln(2),12,16)$

Correct answer:

$(16,12,8\ln(2))$

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

Derivative of an exponential: d[a^u]=a^uln(a)du

Product rule: d[uv]=udv+vdu

Note that may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivatives of f(x,y,z)=x^yy^z at point (1,2,3)

$\frac{\delta f}{\delta x}=x^{y-1}y^{z+1}=1^12^4=16$

$\frac{\delta f}{\delta y}=x^yy^zln(x)+zx^yy^{z-1}=1^22^3ln(1)+3(1^2)2^2=12$

$\frac{\delta f}{\delta z}=x^yy^z\ln(y)=1^22^3\ln(2)=8\ln(2)$

The slope is

$(16,12,8\ln(2))$

Report an Error

Example Question #295 : How To Find Differential Functions

Find the slope of the function $f(x,y)=x^{y^3}$ at the point (2,2)

Possible Answers:

(256,2048)

(2048,4096)

(1719,512)

(512,368)

(1024,2129)

Correct answer:

(1024,2129)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rule will be necessary:

Derivative of an exponential: $d[a^u]=a^u\ln(a)du$

Note that may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivative of $f(x,y)=x^{y^3}$ at the point (2,2)

$\frac{\delta f}{\delta x}=y^3x^{y^3-1}=2^3(2^{2^3-1})=1024$

$\frac{\delta f}{\delta y}=3y^2x^{y^3}ln(x)=3(2^2)(2^{2^3})ln(2)=2129$

The slope is

(1024,2129)

Report an Error

Example Question #296 : How To Find Differential Functions

What is the slope of the function $f(x,y)=y^{\sqrt{x}}$ at the point (4,3) ?

Possible Answers:

(3,3.12)

(1.36,11)

(4.48,9)

(5.89,2)

(2.47,6)

Correct answer:

(2.47,6)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rule will be necessary:

Derivative of an exponential: d[a^u]=a^uln(a)du

Note that may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Find the partial derivatives of the function $f(x,y)=y^{\sqrt{x}}$ at the point (4,3)

$\frac{\delta f}{\delta x}=\frac{1}{2\sqrt{x}}y^{\sqrt{x}}\ln(y)=\frac{1}{2\sqrt{4}}3^{\sqrt{4}}\ln(3)=2.47$

$\frac{\delta f}{\delta y}=\sqrt{x}y^{\sqrt{x}-1}=\sqrt{4}(3^{\sqrt{4}-1})=6$

The slope is

(2.47,6)

Report an Error

Example Question #297 : How To Find Differential Functions

Calculate the slope of the function $f(x,y)=x^{e^y}$ at the point (2,1)

Possible Answers:

(11.11,29.46)

(8.71,52.63)

(2.33,19.77)

(8.94,12.40)

(4.72,6.90)

Correct answer:

(8.94,12.40)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rule will be necessary:

Derivative of an exponentials: $d[e^u]=e^udu;d[a^u]=a^u\ln(a)du$

Note that may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Find the partial derivatives of $f(x,y)=x^{e^y}$ at the point (2,1)

$\frac{\delta f}{\delta x}=e^yx^{e^y-1}=e^1(2^{e^1-1})=8.94$

$\frac{\delta f}{\delta y}=e^yx^{e^y}\ln(x)=e^1(2^{e^1})\ln(2)=12.40$

The slope is

(8.94,12.40)

Report an Error

Example Question #298 : How To Find Differential Functions

Find the slope of the function $f(x,y)=x^{\ln(y)}$ at the point (3,8)

Possible Answers:

(9.98,2.34)

(1.27,10.31)

(6.62,0.73)

(6.81,1.35)

(2,97,6.54)

Correct answer:

(6.81,1.35)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

Derivative of an exponential: $d[a^u]=a^u\ln(a)du$

Derivative of a natural log: $d[\ln(u)]=\frac{du}{u}$

Note that may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Find the partial derivatives of $f(x,y)=x^{\ln(y)}$ at the point (3,8)

$\frac{\delta f}{\delta x}=\ln(y)x^{\ln(y)-1}=\ln(8)3^{\ln(8)-1}=6.81$

$\frac{\delta f}{\delta y}=\frac{x^{\ln(y)}}{y}\ln(x)=\frac{3^{\ln(8)}}{8}\ln(3)=1.35$

The slope is

(6.81,1.35)

Report an Error

Example Question #481 : Functions

What is the slope of the line normal to the function f(x)=x^2+2x at the point x=1 ?

Possible Answers:

$\frac{1}{4}$

$-\frac{1}{4}$

Correct answer:

$-\frac{1}{4}$

Explanation:

The first step to finding the slope of the line normal to a point is to find the slope of the tangent at this point.

The slope of this tangent, in turn, is found by finding the value of the derivative of the function at this point.

Evaluating the function f(x)=x^2+2x at the point x=1

The slope of the tangent is

f'(x)=2x+2;f'(1)=2(1)+2=4

The slope of the normal line is the negative reciprocal of this value. Thus for this problem the normal is

$-\frac{1}{4}$

Report an Error

Example Question #292 : Other Differential Functions

What is the slope of the line normal to the function f(x)=x^3-2x^2+x+5 at the point x=2 ?

Possible Answers:

$\frac{1}{5}$

$-\frac{1}{5}$

$-\frac{2}{7}$

Correct answer:

$-\frac{1}{5}$

Explanation:

The first step to finding the slope of the line normal to a point is to find the slope of the tangent at this point.

The slope of this tangent, in turn, is found by finding the value of the derivative of the function at this point.

Evaluating the function f(x)=x^3-2x^2+x+5 at the point x=2

The slope of the tangent is

f'(x)=3x^2-4x+1;f'(2)=3(2)^2-4(2)+1=5

The slope of the normal line is the negative reciprocal of this value. Thus for this problem the normal is

$-\frac{1}{5}$

Report an Error

View Calculus Tutors

Vandana
Certified Tutor

Jawaharlal Nehru Technological University, Bachelor of Technology, Civil Engineering. Florida International University, Maste...

View Calculus Tutors

Ping
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Computer Science. University of North Texas, Master of Arts, Education.

View Calculus Tutors

Usman
Certified Tutor

New York University, Bachelor in Arts, Political Science and Government.

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

SAT Tutors in Washington DC, SAT Tutors in Los Angeles, Statistics Tutors in Dallas Fort Worth, Physics Tutors in Dallas Fort Worth, Reading Tutors in Los Angeles, Chemistry Tutors in Miami, Spanish Tutors in Atlanta, MCAT Tutors in Denver, Spanish Tutors in Denver, LSAT Tutors in Phoenix

Popular Courses & Classes

SSAT Courses & Classes in Miami, ISEE Courses & Classes in Philadelphia, ACT Courses & Classes in Denver, ACT Courses & Classes in Los Angeles, GRE Courses & Classes in Miami, LSAT Courses & Classes in Seattle, GMAT Courses & Classes in Chicago, ACT Courses & Classes in Dallas Fort Worth, ACT Courses & Classes in Washington DC, SAT Courses & Classes in Houston

Popular Test Prep

GMAT Test Prep in San Francisco-Bay Area, GMAT Test Prep in Chicago, SSAT Test Prep in Dallas Fort Worth, GRE Test Prep in Dallas Fort Worth, LSAT Test Prep in Seattle, ACT Test Prep in Los Angeles, SAT Test Prep in Los Angeles, SSAT Test Prep in Seattle, SAT Test Prep in Philadelphia, ACT Test Prep in Denver

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 1 : How to find differential functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #1501 : Calculus

Example Question #292 : How To Find Differential Functions

Example Question #472 : Functions

Example Question #481 : Differential Functions

Example Question #295 : How To Find Differential Functions

Example Question #296 : How To Find Differential Functions

Example Question #297 : How To Find Differential Functions

Example Question #298 : How To Find Differential Functions

Example Question #481 : Functions

Example Question #292 : Other Differential Functions

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: