We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : 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 #1021 : Functions

Find the first derivative of the following function:

$f(x)=3xe^{x}+12\cos(x)$

Possible Answers:

$3e^x+3xe^x-\sin(x)$

$3e^x+3xe^x-12\sin(x)$

$3e^x+3x^2e^x-12\sin(x)$

$3e^x+3xe^x+12\sin(x)$

$3+3xe^x-12\sin(x)$

Correct answer:

$3e^x+3xe^x-12\sin(x)$

Explanation:

The derivative of the function is equal to

$f'(x)=3e^x+3xe^x-12\sin(x)$

and was found using the following rules:

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

Report an Error

Example Question #1021 : Differential Functions

Find the derivative of the following function at x=1:

$(f\circ g)(x)$

where f(x)=2x^2 , g(x)=5x

Possible Answers:

100

Correct answer:

100

Explanation:

To find the derivative of the composite function, we must rewrite the composition as f(g(x)) . Using the chain rule, the derivative of this function is equal to

$f'(g(x)) \cdot g'(x)$ .

For our function,

f'(x)=4x and g'(x)=5 . These derivatives were found using the following rule:

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

We now have all the pieces to use the chain rule formula, where we are evaluating the function at x=1:

g(1)=5

g'(1)=5

f'(g(1))=f'(5)=20

$20 \cdot 5=100$

Report an Error

Example Question #832 : How To Find Differential Functions

Find the first derivative of the function:

$f(x)= \frac{t^4-t^2}{e^t}$

Possible Answers:

e^t(2t-t^2)

e^t(-t^4+4t^3+t^2-2t)

$\frac{-t^4+4t^3+t^2-2t}{e^{2t}}$

e^t(t^4+4t^3+t^2+2t)

Correct answer:

e^t(-t^4+4t^3+t^2-2t)

Explanation:

The derivative of the function is equal to

$f'(x)=\frac{e^t(4t^3-2t)-e^t(t^4+t^2)}{e^{2t}}=e^t(-t^4+4t^3+t^2-2t)$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} \frac{f(x)}{g(x)}=\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^x=e^x$

Report an Error

Example Question #833 : How To Find Differential Functions

Find the second derivative of the function:

$f(x)=\cos(\sec(e^{10x^2}))$

Possible Answers:

$-20xe^{10x^2}$

$20xe^{10x^2}$

$20xe^{10x^2}\sec(x)\tan(x)$

Correct answer:

$20xe^{10x^2}$

Explanation:

Because $\cos(x)$ and $\sec(x)$ are inverses of each other, we can simplify the given function to

$f(x)=e^{10x^2}$

Taking the derivative of this, we get

$f'(x)=20xe^{10x^2}$

by using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u\frac{\mathrm{du} }{\mathrm{d} x}$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Report an Error

Example Question #1023 : Differential Functions

Determine the slope of the line that is tangent to the function f(x)=sin(10x) at the point $x=\pi$

Possible Answers:

Correct answer:

Explanation:

The slope of the tangent can be found by taking the derivative of the function and evaluating the value of the derivative at a point of interest.

We'll need to make use of the following derivative rule(s):

Trigonometric derivative:

d[sin(u)]=cos(u)du

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

Taking the derivative of the function f(x)=sin(10x) at the point $x=\pi$

The slope of the tangent is

f'(x)=10cos(x)

$f'(\pi)=10cos(\pi)=-10$

Report an Error

Example Question #1021 : Differential Functions

Determine the slope of the line that is tangent to the function $f(x)=sin(5\pi x^2)$ at the point x=2

Possible Answers:

$20\pi$

$-10\pi$

$-20\pi$

$10\pi$

Correct answer:

$20\pi$

Explanation:

The slope of the tangent can be found by taking the derivative of the function and evaluating the value of the derivative at a point of interest.

We'll need to make use of the following derivative rule(s):

Trigonometric derivative:

d[sin(u)]=cos(u)du

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

Taking the derivative of the function $f(x)=sin(5\pi x^2)$ at the point x=2

The slope of the tangent is

$f'(x)=10\pi x cos(5\pi x^2)$

$f'(2)=10\pi (2) cos(5\pi (2)^2)=20\pi$

Report an Error

Example Question #831 : How To Find Differential Functions

Determine the slope of the line that is tangent to the function f(x)=sin^2(5x) at the point $x=\pi$

Possible Answers:

Correct answer:

Explanation:

The slope of the tangent can be found by taking the derivative of the function and evaluating the value of the derivative at a point of interest.

We'll need to make use of the following derivative rule(s):

Trigonometric derivative:

d[sin(u)]=cos(u)du

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

Taking the derivative of the function f(x)=sin^2(5x) at the point $x=\pi$

The slope of the tangent is

f'(x)=10sin(5x)cos(5x)

$f'(\pi)=10sin(5\pi)cos(5\pi)=0$

Report an Error

Example Question #2051 : Calculus

Determine the slope of the line that is tangent to the function f(x)=cos(4x) at the point $x=\frac{\pi}{2}$

Possible Answers:

Correct answer:

Explanation:

The slope of the tangent can be found by taking the derivative of the function and evaluating the value of the derivative at a point of interest.

We'll need to make use of the following derivative rule(s):

Trigonometric derivative:

d[cos(u)]=-sin(u)du

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

Taking the derivative of the function f(x)=cos(4x) at the point $x=\frac{\pi}{2}$

The slope of the tangent is

f'(x)=-4sin(4x)

$f'(\frac{\pi}{2})=-4sin(4(\frac{\pi}{2}))=0$

Report an Error

Example Question #841 : How To Find Differential Functions

Determine the slope of the line that is tangent to the function $f(x)=cos^2(\pi x^2)$ at the point $x=\frac{1}{2}$

Possible Answers:

$-\pi$

$-4\pi$

$-2\pi$

$\pi$

Correct answer:

$-\pi$

Explanation:

The slope of the tangent can be found by taking the derivative of the function and evaluating the value of the derivative at a point of interest.

We'll need to make use of the following derivative rule(s):

Trigonometric derivative:

d[cos(u)]=-sin(u)du

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

Taking the derivative of the function $f(x)=cos^2(\pi x^2)$ at the point $x=\frac{1}{2}$

The slope of the tangent is

$f'(x)=-4\pi x sin(\pi x^2)cos(\pi x^2)$

$f'(\frac{1}{2})=-4\pi (\frac{1}{2}) sin(\pi (\frac{1}{2})^2)cos(\pi (\frac{1}{2})^2)=-\pi$

Report an Error

Example Question #842 : How To Find Differential Functions

Determine the slope of the line that is tangent to the function f(x)=cos(sin(x)) at the point $x=3\pi$

Possible Answers:

Correct answer:

Explanation:

The slope of the tangent can be found by taking the derivative of the function and evaluating the value of the derivative at a point of interest.

We'll need to make use of the following derivative rule(s):

Trigonometric derivative:

d[sin(u)]=cos(u)du

d[cos(u)]=-sin(u)du

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

Taking the derivative of the function f(x)=cos(sin(x)) at the point $x=3\pi$

The slope of the tangent is

f'(x)=cos(x)sin(sin(x))

$f'(3\pi)=cos(3\pi)sin(sin(3\pi))=0$

Report an Error

View Calculus Tutors

Shane
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Computer Engineering, General.

View Calculus Tutors

Chernyeh
Certified Tutor

Purdue University-Main Campus, Bachelor of Science, Business Administration and Management.

View Calculus Tutors

Jacob
Certified Tutor

Ohio State University-Main Campus, Bachelor of Science, Political Science and Government.

All Calculus 1 Resources

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

Popular Subjects

Calculus Tutors in Washington DC, SSAT Tutors in Phoenix, Math Tutors in Seattle, Chemistry Tutors in Dallas Fort Worth, Physics Tutors in San Francisco-Bay Area, Math Tutors in Phoenix, ISEE Tutors in Boston, Algebra Tutors in San Diego, Algebra Tutors in Chicago, French Tutors in Houston

Popular Courses & Classes

SAT Courses & Classes in Philadelphia, GMAT Courses & Classes in Phoenix, GRE Courses & Classes in Boston, GMAT Courses & Classes in Philadelphia, SSAT Courses & Classes in Philadelphia, GMAT Courses & Classes in Chicago, MCAT Courses & Classes in Washington DC, ISEE Courses & Classes in San Francisco-Bay Area, GRE Courses & Classes in Los Angeles, GRE Courses & Classes in Miami

Popular Test Prep

GMAT Test Prep in Chicago, GRE Test Prep in New York City, GRE Test Prep in Los Angeles, LSAT Test Prep in Washington DC, SAT Test Prep in Chicago, GMAT Test Prep in Phoenix, SSAT Test Prep in Phoenix, GRE Test Prep in Houston, ISEE Test Prep in Atlanta, LSAT 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 1 : Differential Functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #1021 : Functions

Example Question #1021 : Differential Functions

Example Question #832 : How To Find Differential Functions

Example Question #833 : How To Find Differential Functions

Example Question #1023 : Differential Functions

Example Question #1021 : Differential Functions

Example Question #831 : How To Find Differential Functions

Example Question #2051 : Calculus

Example Question #841 : How To Find Differential Functions

Example Question #842 : How To Find Differential Functions

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: