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 #1491 : Calculus

Find the slope of the tangent line at x=11 .

$x^{3}(x+3)$

Possible Answers:

1893

1794

1894

6413

Correct answer:

6413

Explanation:

Begin by finding the derivative using the product rule.

The product rule is,

f(x)g(x)'=f(x)g'(x)+g(x)f'(x) .

Applying this rule to our function where

$f(x)=x^3\rightarrow f'(x)=3x^2$

$g(x)=x+3\rightarrow g'(x)=1$

we get,

x^3(1)+(x+3)3x^2

x^3+3x^3+9x^2

4x^3+9x^2

Now, substitute 11 for x.

$4(11)^{3}+9\cdot11^{2}$

=5324+1089

=6413

Report an Error

Example Question #461 : Functions

Find the slope of the tangent line at x=1 .

$(x+3)x^{4}$

Possible Answers:

Correct answer:

Explanation:

Begin by finding the derivative using the product rule.

The product rule is,

f(x)g(x)'=f(x)g'(x)+g(x)f'(x) .

Applying this rule to our function where

$f(x)=(x+3)\rightarrow f'(x)=1$

$g(x)=x^4\rightarrow g'(x)=4x^3$

we get,

$(x+3)4x^3+x^{4}(1)$

$4x^{4}+12x^{3}+x^{4}$

$5x^{4}+12x^{3}$

Now, substitute 1 for x.

Report an Error

Example Question #461 : Functions

What is the slope of the tanget line at x=2.5 ?

$2x^{4}-3x^{2}+2$

Possible Answers:

110

115

105

100

Correct answer:

110

Explanation:

First, find the derivative using the power rule which states, $(x^n)'=nx^{n-1}$ .

Applying the power rule to each term in the function we get,

$8x^{3}-6x$

Now, plug in 2.5 for x.

$8(2.5^{3})-6(2.5)$

125-15=110

Report an Error

Example Question #281 : Other Differential Functions

Find the slope of the tangent line at x=1.5 .

$\frac{x^{2}}{4}$

Possible Answers:

$\frac{1}{2}$

$\frac{3}{4}$

$\frac{1}{4}$

Correct answer:

$\frac{3}{4}$

Explanation:

First, find the derivative using the quotient rule which states,

$\frac{f(x)}{g(x)}'=\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$ .

In this particular case,

$f(x)=x^2\rightarrow f'(x)=2x$

$g(x)=4\rightarrow g'(x)=0$ .

Therefore the derivative becomes,

$\frac{4(2x)-x^2(0)}{4^2}=\frac{4(2x)}{16}$

$=\frac{8x}{16}=\frac{x}{2}$

Now, substitute 1.5 for x.

$\frac{1.5}{2}=\frac{3}{4}$

Report an Error

Example Question #281 : Other Differential Functions

Find the derivative.

$\frac{4}{2x}$

Possible Answers:

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

$-x^{2}$

$-\frac{2}{x^{3}}$

Correct answer:

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

Explanation:

Use the quotient rule to find the derivative which states,

$\frac{f(x)}{g(x)}'=\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$ .

In this particular case,

$f(x)=4\rightarrow f'(x)=0$

$g(x)=2x\rightarrow g'(x)=2$ .

Therefore the derivative becomes,

$\frac{2x(0)-4(2)}{(2x)^2}=\frac{-8}{4x^{2}}$

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

Report an Error

Example Question #1501 : Calculus

Find the derivative.

$\frac{2x}{x^{3}}$

Possible Answers:

$\frac{-4}{x^{4}}$

$\frac{-4}{x^3}$

$\frac{-3}{x^{3}}$

$\frac{-2}{x^5}$

Correct answer:

$\frac{-4}{x^3}$

Explanation:

Use the quotient rule to find the derivative which states,

$\frac{f(x)}{g(x)}'=\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$ .

In this particular case,

$f(x)=2x\rightarrow f'(x)=2$

$g(x)=x^3\rightarrow g'(x)=3x^2$ .

Therefore, the derivative becomes,

$=\frac{2x^{3}-2x(3x^{2})}{x^{6}}$

$=\frac{-4x^{3}}{x^6}$

$=\frac{-4}{x^3}$

Report an Error

Example Question #287 : Other Differential Functions

What is the slope of the function f(x,y)=x^y at (e,4) ?

Possible Answers:

$(4e^4,e^4\ln(4))$

(4e^3,e^4)

$(4e^3,e^4\ln(4))$

(e^3,e^4)

(e^4,e^4)

Correct answer:

(4e^3,e^4)

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 derivatives of f(x,y)=x^y at (e,4)

$\frac{\delta f}{\delta x}=yx^{y-1}=4e^3$

$\frac{\delta f}{\delta y}=x^y\ln(x)=e^4\ln(e)=e^4$

The slope is

(4e^3,e^4)

Report an Error

Example Question #288 : Other Differential Functions

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

Possible Answers:

(305.4,866.9)

(457.2,203.3)

(203.3,457.2)

(866.9,305.4)

Correct answer:

(866.9,305.4)

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: $d[a^u]=a^u\ln(a)du$

Product rule: d[uv]=udv+vdu

Note that and 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^yy^x at the point (2,4)

$\frac{\delta f}{\delta x}=x^{y-1}y^{x+1}+x^{y}y^{x}ln(y)=2^34^3+2^44^2ln(4)=866.9$

$\frac{\delta f}{\delta y}=x^{y}y^{x}ln(x)+x^{y+1}y^{x-1}=2^44^2ln(2)+2^54=305.4$

The slope is

(866.9,305.4)

Report an Error

Example Question #289 : Other Differential Functions

Find the slope of the function $f(x,y)=x^{2+2y}$ at (1,7) .

Possible Answers:

(2,2)

(0,16)

(4,8)

(8,4)

(16,0)

Correct answer:

(16,0)

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: $d[a^u]=a^u\ln(a)du$

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 $f(x,y)=x^{2+2y}$ at (1,7)

$\frac{\delta f}{\delta x}=(2+2y)x^{1+2y}=(2+14)1^{15}=16$

$\frac{\delta f}{\delta y}=2x^{2+2y}\ln(x)=2(1^{16})\ln(1)=0$

The slope is

(16,0)

Report an Error

Example Question #290 : Other Differential Functions

Find the slope of the function $f(x,y,z)=x^{yz^2}$ at the point (2,3,1)

Possible Answers:

(5.5,33.3,12)

(12,5.5,33.3)

(33.3,5.5,12)

(12,33.3,5.5)

(5.5,12,33.3)

Correct answer:

(12,5.5,33.3)

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 $f(x,y,z)=x^{yz^2}$ at the point (2,3,1)

$\frac{\delta f}{\delta x}=yz^2x^{yz^2-1}=3(1^2)2^{3(1^2)-1}=12$

$\frac{\delta f}{\delta y}=z^2x^{yz^2}\ln(x)=1^2(2^{3(1^2)})\ln(2)=5.5$

$\frac{\delta f}{\delta z}=2yzx^{yz^2}\ln(x)=2(3)(1)(2^{3(1^2)})\ln(2)=33.3$

The slope is

(12,5.5,33.3)

Report an Error

View Calculus Tutors

Juhyeon
Certified Tutor

Vanderbilt University, Master of Science, Environmental Engineering.

View Calculus Tutors

Charles
Certified Tutor

Pennsylvania State University-Main Campus, Bachelor of Science, Mathematics.

View Calculus Tutors

Cynthia
Certified Tutor

North Carolina A & T State University, Bachelor of Science, Mechanical Engineering.

All Calculus 1 Resources

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

Popular Subjects

ISEE Tutors in Phoenix, GRE Tutors in San Francisco-Bay Area, Statistics Tutors in San Diego, Biology Tutors in San Francisco-Bay Area, English Tutors in San Francisco-Bay Area, French Tutors in Atlanta, GMAT Tutors in Boston, Chemistry Tutors in Atlanta, GMAT Tutors in Seattle, SSAT Tutors in Denver

Popular Courses & Classes

ACT Courses & Classes in Chicago, GMAT Courses & Classes in Washington DC, Spanish Courses & Classes in San Diego, MCAT Courses & Classes in Chicago, ACT Courses & Classes in Philadelphia, LSAT Courses & Classes in Phoenix, SSAT Courses & Classes in Philadelphia, ISEE Courses & Classes in Atlanta, ACT Courses & Classes in San Diego, ISEE Courses & Classes in Denver

Popular Test Prep

SAT Test Prep in Seattle, ACT Test Prep in Philadelphia, ACT Test Prep in Chicago, MCAT Test Prep in Philadelphia, SAT Test Prep in San Diego, SAT Test Prep in Boston, MCAT Test Prep in Chicago, GMAT Test Prep in Philadelphia, SAT Test Prep in Philadelphia, SAT 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 #1491 : Calculus

Example Question #461 : Functions

Example Question #461 : Functions

Example Question #281 : Other Differential Functions

Example Question #281 : Other Differential Functions

Example Question #1501 : Calculus

Example Question #287 : Other Differential Functions

Example Question #288 : Other Differential Functions

Example Question #289 : Other Differential Functions

Example Question #290 : 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: