Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Derivative at a Point

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

← Previous 1 2 3 4 5 6 7 8 9 10 11 12 Next →

Example Question #71 : Derivative At A Point

Find f'(0) for

$f(x)=\sin(x)\ln(x+1)$

Possible Answers:

$\frac{\pi}{4}$

$\frac{\pi}{2}$

$\pi$

Correct answer:

Explanation:

In order to take the derivative, we need to remember how to take derivative of natural log and the product rule.

Derivative of natural log:

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

$f'(t)=\frac{g'(t)}{g(t)}$

Product Rule:

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

Now lets apply both of these rule to our problem.

$f(x)=\sin(x)\ln(x+1)$

$f'(x)=\cos(x)\ln(x+1)+\frac{\sin(x)}{x+1}$

We plug in 0 to get.

$f'(0)=\cos(0)\ln(1)+\frac{\sin(0)}{1}$

$f'(0)=1*0+\frac{0}{1}=0$

Report an Error

Example Question #72 : Derivative At A Point

Find f'(1) for

$f(x)=x^2\cos(x)$

Possible Answers:

$2\cos(1)$

The derivative doesn't exist.

$2\cos(1)-\sin(1)$

$2\tan(1)$

$2\cos(1)+\sin(1)$

Correct answer:

$2\cos(1)-\sin(1)$

Explanation:

In order to find f'(1) , we first find f'(x) .

To find the derivative, we need to remember the product rule.

Product Rule:

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

Lets apply this rule to our problem.

$f(x)=x^2\cos(x)$

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

We plug in 1 to get

$f'(1)=2(1)\cos(1)-(1)^2\sin(1)$

$=2\cos(1)-\sin(1)$

Report an Error

Example Question #73 : Derivative At A Point

Find f'(0) for

$f(x)=2^{2x+2}$

Possible Answers:

$\log(2)$

$8\ln(2)$

Correct answer:

$8\ln(2)$

Explanation:

In order to find f'(0) , we need to find f'(x) .

We need to remember the derivative of expotential functions.

Derivative of expotential functions:

$f(x)=a^{g(x)}$

$f'(x)=g'(x)a^{g(x)}\ln(a)$

Now lets apply this to our problem.

$f(x)=2^{2x+2}$

$f'(x)=2*2^{2x+2}*\ln(2)$

$f'(0)=2*2^{2(0)+2}*\ln(2)=8\ln(2)$

Report an Error

Example Question #74 : Derivative At A Point

Find f'(0) for

$f(t)=t^3+2^{26t^2}$

Possible Answers:

$\log(2)$

Correct answer:

Explanation:

In order to take the derivative, we need to remember how to take derivatives of exponential functions not in base e.

Derivatives of exponential functions not in base e:

$f(x)=a^{g(x)}$

$f'(x)=g'(x)a^{g(x)}\ln(a)$

Now lets apply this to our problem.

$f(t)=t^3+2^{26t^2}$

$f'(t)=3t^2+52t\ 2^{26t^2}\ln(2)$

Now we plug in 0 to get.

$f'(0)=3(0)^2+52(0) \2^{26(0)^2}\ln(2)$

$f'(0)=0+0*1*\log(2)=0$

Report an Error

Example Question #75 : Derivative At A Point

What is the slope of $f(x)=-x^{2}+10x-5$ at (-5,3) ?

Possible Answers:

Correct answer:

Explanation:

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

Since we have $f(x)=-x^{2}+10x-5$ , we can use the Power Rule

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

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

We also have a point (-5,3) with a -coordinate , so the slope

f'(-5)=-2(-5)+10=10+10=20 .

Report an Error

Example Question #76 : Derivative At A Point

What is the slope of $f(x)=7x^{2}-2x+9$ at (2,0) ?

Possible Answers:

Correct answer:

Explanation:

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

Since we have

$f(x)=7x^{2}-2x+9$ , we can use the Power Rule

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

to determine that

$f'(x)=(2)7x^{2-1}-(1)2x^{1-1}+(0)9=14x-2$ .

We also have a point (2,0) with a -coordinate , so the slope

f'(2)=14(2)-2=28-2=26 .

Report an Error

Example Question #77 : Derivative At A Point

What is the slope of $f(x)=6x^{2}+7x+1$ at (-1,4) ?

Possible Answers:

Correct answer:

Explanation:

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

Since we have

$f(x)=6x^{2}+7x+1$ , we can use the Power Rule

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

to determine that

$f'(x)=(2)6x^{2-1}+(1)7x^{1-1}+(0)1=12x+7$ .

We also have a point (-1,4) with a -coordinate , so the slope

f'(-1)=12(-1)+7=-12+7=-5 .

Report an Error

Example Question #78 : Derivative At A Point

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

$f(x)=2xe^{\cos(x)}$

Possible Answers:

4e^1

-2e^1

2e^1

Correct answer:

2e^1

Explanation:

The derivative of the function is

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

and was found using the following rules:

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

Now, just plug zero into the first derivative function to get our answer:

f'(0)=2e^1

Report an Error

Example Question #79 : Derivative At A Point

Evaluate the derivative of the following function at x=0 :

$f(x)=5x^4e^{\sin(x)}$

Possible Answers:

Correct answer:

Explanation:

The derivative of the function is

$f'(x)=20x^3e^{\sin(x)}+5x^4\cos(x)e^{\sin(x)}$

and was found using the following rules:

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

To finish, plug in the point given into the first derivative function:

f'(0)=0+0=0

Report an Error

Example Question #80 : Derivative At A Point

Find the derivative of the following function at x=2 :

$f(x)=x^2+\csc(2x)+e^\ln(x)$

Possible Answers:

$2x+\csc(2x)+\cot(2x)$

$5-2\csc(4)\cot(4)$

$6+\csc(2x)$

$5+2\csc(x)\cot(x)$

Correct answer:

$5-2\csc(4)\cot(4)$

Explanation:

The derivative of the function is:

$f'(x)=2x-2\csc(2x)\cot(2x)+1$

and was found using the following rules:

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

Now, just plug in the point x=2 into the first derivative function:

$f'(2)=5-2\csc(4)\cot(4)$

Report an Error

← Previous 1 2 3 4 5 6 7 8 9 10 11 12 Next →

View Calculus 2 Tutors

Yoonsik
Certified Tutor

Seoul National University, Bachelor of Science, Physics. University of Pennsylvania, Doctor of Philosophy, Atomic and Molecul...

View Calculus 2 Tutors

Kurosh
Certified Tutor

University of tehran, Bachelor of Engineering, Electrical Engineering.

View Calculus 2 Tutors

Filiberto
Certified Tutor

University of Arizona, Bachelor of Science, Electrical Engineering.

All Calculus 2 Resources

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

Popular Subjects

ISEE Tutors in Boston, MCAT Tutors in San Diego, Physics Tutors in Houston, LSAT Tutors in Houston, Reading Tutors in Miami, Physics Tutors in Chicago, English Tutors in Atlanta, French Tutors in Dallas Fort Worth, Biology Tutors in Los Angeles, ISEE Tutors in Miami

Popular Courses & Classes

SSAT Courses & Classes in Denver, MCAT Courses & Classes in Los Angeles, SAT Courses & Classes in Chicago, Spanish Courses & Classes in New York City, GRE Courses & Classes in Philadelphia, LSAT Courses & Classes in San Francisco-Bay Area, LSAT Courses & Classes in Dallas Fort Worth, SSAT Courses & Classes in New York City, GRE Courses & Classes in New York City, GMAT Courses & Classes in Los Angeles

Popular Test Prep

ISEE Test Prep in Seattle, GRE Test Prep in Phoenix, MCAT Test Prep in Phoenix, ISEE Test Prep in Los Angeles, GMAT Test Prep in Atlanta, GMAT Test Prep in Washington DC, ISEE Test Prep in Phoenix, ACT Test Prep in San Diego, GMAT Test Prep in Los Angeles, MCAT Test Prep in New York City

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 at a Point

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #71 : Derivative At A Point

Example Question #72 : Derivative At A Point

Example Question #73 : Derivative At A Point

Example Question #74 : Derivative At A Point

Example Question #75 : Derivative At A Point

Example Question #76 : Derivative At A Point

Example Question #77 : Derivative At A Point

Example Question #78 : Derivative At A Point

Example Question #79 : Derivative At A Point

Example Question #80 : Derivative At A Point

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: