We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Derivative Review

Study concepts, example questions & explanations for Calculus 2

Create An Account

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #131 : Derivative Review

What is the slope of a function $f(x)=3x^{2}+7x-5$ at (2,6) ?

Possible Answers:

Correct answer:

Explanation:

We define slope as the first derivative of a given function.

Since we have

$f(x)=3x^{2}+7x-5$ , we can use the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ to determine that

$f'(x)=(2)3x^{2-1}+(1)7x^{1-1}-(0)5=6x+7$ .

We also have a point (2,6) with a -coordinate , so the slope

f'(2)=6(2)+7=12+7=19 .

Report an Error

Example Question #41 : Derivative At A Point

What is the slope of a function $f(x)=5x^{2}-9x+14$ at (-4,2) ?

Possible Answers:

Correct answer:

Explanation:

We define slope as the first derivative of a given function.

Since we have

$f(x)=5x^{2}-9x+14$ , we can use the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ to determine that

$f'(x)=(2)5x^{2-1}-(1)9x^{1-1}+(0)14=10x-9$ .

We also have a point (-4,2) with a -coordinate , so the slope

f'(-4)=10(-4)-9=-40-9=-49 .

Report an Error

Example Question #133 : Derivative Review

What is the slope of a function $f(x)=2x^{2}+13x-7$ at (-1,1) ?

Possible Answers:

Correct answer:

Explanation:

We define slope as the first derivative of a given function.

Since we have

$f(x)=2x^{2}+13x-7$ , we can use the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ to determine that

$f'(x)=(2)2x^{2-1}+(1)13x^{1-1}-(0)7=4x+13$ .

We also have a point (-1,1) with a -coordinate , so the slope

f'(-1)=4(-1)+13=-4+13=9 .

Report an Error

Example Question #11 : Derivative Defined As Limit Of Difference Quotient

Find f'(0) for

$f(x)=x^3\ln(x+1)$

Possible Answers:

$\infty$

Correct answer:

Explanation:

In order to find the derivative, we need to find f'(x) . We can find this by remembering the product rule and knowing the derivative of natural log.

Product Rule:

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

Derivative of natural log:

$f(x)=\ln(g(x))$

$f'(x)=\frac{g'(x)}{g(x)}$

Now lets apply this to our problem.

$f(x)=x^3\ln(x+1)$

$f'(x)=3x^2\ln(x+1)+\frac{x^3}{x+1}$

$f'(0)=3(0)^2\ln(0+1)+\frac{0^3}{0+1}=0$

Report an Error

Example Question #51 : Derivative At A Point

Find the derivative of the following function at the point z=0 :
$f(z)=\frac{z^2\cos(z)}{2}$

Possible Answers:

Correct answer:

Explanation:

The derivative of the function is

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

which was found using the following rules:

$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x)f'(x)-f(x)g'(x)}{(g(x))^2}$ ,

$\frac{d}{dx}\cos(x)=-\sin(x)$ ,

$\frac{d}{dx}(x^n)=nx^{n-1}$

Finally, plug in the point z=0 into the first derivative function:

$f'(0)=\frac{0}{4}=0$

Report an Error

Example Question #52 : Derivative At A Point

Find the derivative of the following function at $\theta=2\pi$ :

$f(\theta )=\sin^3(\theta)+e^{\theta}$

Possible Answers:

$e^{2\pi}$

$3+2\pi e^{2\pi}$

Correct answer:

$e^{2\pi}$

Explanation:

The derivative of the function is

$f'(\theta)=3\sin^2(\theta)\cos(\theta)+ e^{\theta}$

and was found using the following rules:

$\frac{d}{dx}(e^{u})=e^{u}\frac{du}{dx}$ ,

$\frac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)$ ,

$\frac{d}{dx}\sin(x)=\cos(x)$ ,

$\frac{d}{dx}(x^n)=nx^{^{n-1}}$

To finish the problem, plug in $\theta=2 \pi$ into the derivative function:

$f'(2 \pi)= e^{2\pi}$ .

Report an Error

Example Question #53 : Derivative At A Point

Find the derivative of the following function at $y=2 \pi$

$f(y)=\sec^2(y)+\tan(y)$ .

Possible Answers:

undefined

$\infty$

Correct answer:

Explanation:

The derivative of the function is

$f'(y)=2\sec^2(y)\tan(y)+\sec^2(y)$

and was found using the following rules:

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

Then, plug in the point given into the first derivative function:

$f'(2 \pi) = 2(1)(0)+1=1$

Report an Error

Example Question #54 : Derivative At A Point

Find the second derivative of the following function at x=0 :

$f(x)=\cos(x)+e^\frac{x}{2}+x^3+x^2$

Possible Answers:

$\frac{5}{4}$

$3\tfrac{1}{4}$

$2\tfrac{1}{4}$

Correct answer:

$\frac{5}{4}$

Explanation:

We must find the first derivative of the function first:

$f'(x)=-\sin(x)+\frac{e^{\frac{x}{2}}}{2}+3x^2+2x$

The derivative was found using the following rules:

$\frac{d}{dx}\cos(x)=-\sin(x)$ , $\frac{d}{dx}(e^u)=e^u\frac{du}{dx}$ , $\frac{d}{dx}(x^n)=nx^{n-1}$

Find the second derivative of the function by taking the derivative of the above function:

$f''(x)=-\cos(x)+\frac{e^{\frac{x}{2}}}{4}+6x+2$

An additional rule was used:

$\frac{d}{dx}\sin(x)=\cos(x)$

Now, plug in x=0 into the above function:

$f''(0)=-1+\frac{1}{4}+2=\frac{5}{4}$

Report an Error

Example Question #51 : Derivative At A Point

Find the second derivative of the following function at x=0 :

$e^{2x}+2x\cos(3x)$

Possible Answers:

Correct answer:

Explanation:

To find the second derivative of the function, we first must find the first derivative of the function:

$f'(x)=2e^{2x}+2\cos(3x)-6x\sin(3x)$

The derivative was found using the following rules:

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

The second derivative is simply the derivative of the first derivative function, and is equal to:

$f''(x)=4e^{2x}-12\sin(3x)-18x\cos(3x)$

One more rule used in combination with some of the ones above is:

$\frac{d}{dx}\sin(x)=\cos(x)$

To finish the problem, plug in x=0 into the above function to get an answer of .

Report an Error

Example Question #140 : Derivative Review

What is the slope of $f(x)=9x^{2}-12x+5$ at (4,-4) ?

Possible Answers:

Correct answer:

Explanation:

We define slope as the first derivative of a given function.

Since we have

$f(x)=9x^{2}-12x+5$ , we can use the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ to determine that

$f'(x)=(2)9x^{2-1}-(1)12x+(0)5=18x-12$ .

We also have a point (4,-4) with a -coordinate , so the slope

f'(4)=18(4)-12=72-12=60 .

Report an Error

View Calculus 2 Tutors

Shawn
Certified Tutor

Old Dominion University, Bachelor of Science, Mathematics. Old Dominion University, Master of Science, Computational Mathemat...

View Calculus 2 Tutors

Gregory
Certified Tutor

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

View Calculus 2 Tutors

Mohammad
Certified Tutor

Rutgers University-New Brunswick, Bachelor of Science, Mathematics. Rutgers University-New Brunswick, Master of Science, Stat...

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

GMAT Tutors in Boston, GMAT Tutors in Houston, Reading Tutors in Philadelphia, Chemistry Tutors in Dallas Fort Worth, LSAT Tutors in Philadelphia, Spanish Tutors in Washington DC, French Tutors in Philadelphia, Math Tutors in San Diego, Chemistry Tutors in Atlanta, SSAT Tutors in Seattle

Popular Courses & Classes

ACT Courses & Classes in San Francisco-Bay Area, LSAT Courses & Classes in Atlanta, SSAT Courses & Classes in Atlanta, SSAT Courses & Classes in Chicago, GRE Courses & Classes in Boston, MCAT Courses & Classes in Atlanta, GMAT Courses & Classes in Houston, LSAT Courses & Classes in Miami, SSAT Courses & Classes in Boston, ISEE Courses & Classes in Los Angeles

Popular Test Prep

GRE Test Prep in Atlanta, ACT Test Prep in Dallas Fort Worth, MCAT Test Prep in Dallas Fort Worth, SAT Test Prep in San Diego, LSAT Test Prep in Dallas Fort Worth, ACT Test Prep in San Francisco-Bay Area, ISEE Test Prep in Atlanta, LSAT Test Prep in New York City, ISEE Test Prep in San Diego, GMAT Test Prep in Philadelphia

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 2 : Derivative Review

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #131 : Derivative Review

Example Question #41 : Derivative At A Point

Example Question #133 : Derivative Review

Example Question #11 : Derivative Defined As Limit Of Difference Quotient

Example Question #51 : Derivative At A Point

Example Question #52 : Derivative At A Point

Example Question #53 : Derivative At A Point

Example Question #54 : Derivative At A Point

Example Question #51 : Derivative At A Point

Example Question #140 : Derivative Review

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: