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 #181 : Derivative Review

Calculate the derivative of f(x)=2sin(4x)+3sin(10x)+5sin(3x)+11sin(6x) at the point $x=\frac{\pi}{2}$ .

Possible Answers:

Correct answer:

Explanation:

There are 2 steps to solving this problem.

First, take the derivative of f(x) .

Then, replace the value of x with the given point and evaluate.

For example, if x=a , then we are looking for the value of f'(x=a) , or the derivative of f(x) at x=a .

f(x)=2sin(4x)+3sin(10x)+5sin(3x)+11sin(6x)

Calculate f'(x) .

Derivative rules that will be needed here:

$\frac{\mathrm{d} }{\mathrm{d} t} sin(kt) = kcos(kt)$

f'(x)=8cos(4x)+30cos(10x)+15cos(3x)+66cos(6x)

Then, plug in the value of x and evaluate.

$\newline f'(x=\frac{\pi}{2})=8cos(2\pi)+30cos(5\pi)+15cos(\frac{3\pi}{2})+66cos(3\pi) = 8*1+30*-1+15*0+66*-1=8-30-66=-88$

Report an Error

Example Question #181 : Derivative Review

Calculate the derivative of f(x)=cos(3x)+sin(x) at the point $x=\pi$ .

Possible Answers:

$\pi$

Correct answer:

Explanation:

There are 2 steps to solving this problem.

First, take the derivative of f(x) .

Then, replace the value of x with the given point and evaluate.

For example, if x=a , then we are looking for the value of f'(x=a) , or the derivative of f(x) at x=a .

f(x)=cos(3x)+sin(x)

Calculate f'(x)

Derivative rules that will be needed here:

$\frac{\mathrm{d} }{\mathrm{d} t} cos(kt) = -ksin(kt)$
$\frac{\mathrm{d} }{\mathrm{d} t} sin(kt) = kcos(kt)$

f'(x)=-1*3sin(3x)+cos(x)

Then, plug in the value of x and evaluate.

$f'(x=\pi)=-3sin(3\pi)+cos(\pi) = -3*0-1 = -1$

Report an Error

Example Question #101 : Derivative At A Point

Calculate the derivative of $f(x)=cos(5x)+ 3e^{3x}+6x^3$ at the point x=3 .

Possible Answers:

-5sin(15)+9e^9+162

-5sin(5)+9e^3+162

-5sin(15)+e^9+162

Correct answer:

-5sin(15)+9e^9+162

Explanation:

There are 2 steps to solving this problem.

First, take the derivative of f(x) .

Then, replace the value of x with the given point and evaluate.

For example, if x=a , then we are looking for the value of f'(x=a) , or the derivative of f(x) at x=a .

$f(x)=cos(5x)+ 3e^{3x}+6x^3$

Calculate f'(x)

Derivative rules that will be needed here:

Taking a derivative on a term, or using the power rule, can be done by doing the following: $\frac{\mathrm{d} }{\mathrm{d} t} t^n = n * t^{n-1}$
Special rule when differentiating an exponential function: $\frac{\mathrm{d} }{\mathrm{d} t} e^{kt} = ke^{kt}$ , where k is a constant.
$\frac{\mathrm{d} }{\mathrm{d} t} cos(kt) = -ksin(kt)$

$f'(x)=-5sin(5x)+9e^{3x}+18x^2$

Then, plug in the value of x and evaluate.

$\newline f'(x=3)=-5sin(5*3)+9e^{3*3}+18*3^2=-5sin(15)+9e^9+18*9 = -5sin(15)+9e^9+162$

Report an Error

Example Question #182 : Derivative Review

Determine the derivative of the following function at x=2 .

$e^{x^2-3x+2}-ln{x^2}$

Possible Answers:

e^3-1

e^3

Correct answer:

Explanation:

For this function we will need to use the power rule, the exponential rule, and the chain rule.

Power Rule: $x^n \rightarrow nx^{n-1}$

Exponential Rule: $e^u \rightarrow e^u\cdot \frac{du}{dx}$

Chain Rule: $[f(g(x))]'\rightarrow f'(g(x))\cdot g'(x)$

Applying these rules to our function we get the following derivative.

$\frac{\mathrm{d} }{\mathrm{d} x}(e^{x^2-3x+2}-ln{x^2})= (2x-3)e^{x^2-3x+2}-\frac{2}{x}$

Now, plug in x=2 to solve,

$(2*2-3)e^{2^2-3*2+2}-\frac{2}{2}=1-1=0$ .

Report an Error

Example Question #183 : Derivative Review

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

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

Possible Answers:

$\frac{1}{3}$

$-\frac{1}{3}$

Correct answer:

$\frac{1}{3}$

Explanation:

The derivative of the function is

$f'(x)=\frac{1}{3}(e^x\cos(x)-e^x\sin(x))$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ ,

$\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u\frac{\mathrm{du} }{\mathrm{d} x}$ ,

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

Evaluated at the point x=0, we get

$f'(0)=\frac{1}{3}(1-0)=\frac{1}{3}$ .

Report an Error

Example Question #1311 : Calculus Ii

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

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

Possible Answers:

Correct answer:

Explanation:

The derivative of the function is

$f'(x)=\frac{(3x+1)(6x^2+2x+e^x)-3(2x^3+x^2+e^x)}{(3x+1)^2}$

and was found using the following rules:

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

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

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

$f'(x)=\frac{(3(0)+1)(6(0)^2+2(0)+e^{(0)})-3(2(0)^3+(0)^2+e^{(0)})}{(3(0)+1)^2}$

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

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

Report an Error

Example Question #181 : Derivative Review

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

$f(x)=\tan(x)-\frac{3x}{4}$

Possible Answers:

$\frac{5}{4}$

$\frac{1}{4}$

$\frac{3}{4}$

Correct answer:

$\frac{1}{4}$

Explanation:

The derivative of the function is

$f'(x)=\sec^2(x)-\frac{3}{4}$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} \tan(x)=\sec^2(x)$ ,

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

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

$f'(0)=1-\frac{3}{4}=\frac{1}{4}$

Report an Error

Example Question #186 : Derivative Review

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

$f(x)=x^3\cos(x)+\sin(3x)$

Possible Answers:

Correct answer:

Explanation:

The derivative of the function is

$f'(x)=3x^2\cos(x)-x^3\sin(x)+3\cos(3x)$

and was found using the following rules:

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

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

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

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

Now, plug in the point we want into the derivative and solve:

f'(0)=0-0+3(1)=3

Report an Error

Example Question #187 : Derivative Review

Evaluate the derivative at the point $\theta=\pi$ :

$f(\theta)=\cos^2(\theta)+e^\theta$

Possible Answers:

$2\pi$

$e^\pi$

Correct answer:

$e^\pi$

Explanation:

The derivative of the function is

$f'(\theta)=-2\cos(\theta)\sin(\theta)+e^\theta$

and was found using the following rules:

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

$\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u\frac{\mathrm{du} }{\mathrm{d} x}$ ,

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

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

Now, plug in the point asked into the above function and solve:

$f'(\pi)=0+e^\pi=e^\pi$

Report an Error

Example Question #182 : Derivative Review

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

$f(\theta)=\tan(\theta)+\theta e^\theta$

Possible Answers:

$e^\pi (\pi+1)-1$

$e^\pi (\pi-1)$

$e^\pi (\pi+1)+1$

$e^\pi (\pi+1)$

Correct answer:

$e^\pi (\pi+1)-1$

Explanation:

The derivative of the function is

$f'(\theta)=\sec^2(\theta)+e^\theta+\theta e^\theta$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u\frac{\mathrm{du} }{\mathrm{d} x}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \tan(x)=\sec^2(x)$

Finally, plug in $\pi$ for $\theta$ and we get our final answer:

$f'(\theta)=e^\pi (\pi+1)-1$

Report an Error

View Calculus 2 Tutors

Priyanka
Certified Tutor

CUNY Hunter College, Bachelor in Arts, Economics.

View Calculus 2 Tutors

Darryl
Certified Tutor

University of Florida, Bachelor of Science, Mathematics.

View Calculus 2 Tutors

Mohammed
Certified Tutor

University of Maryland-College Park, Bachelor of Science, Electrical Engineering. University of Maryland-College Park, Master...

All Calculus 2 Resources

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

Popular Subjects

GRE Tutors in Seattle, Spanish Tutors in San Francisco-Bay Area, ISEE Tutors in Atlanta, ISEE Tutors in Chicago, Calculus Tutors in Seattle, GRE Tutors in Atlanta, GRE Tutors in New York City, SSAT Tutors in Atlanta, Math Tutors in Los Angeles, Biology Tutors in Chicago

Popular Courses & Classes

ISEE Courses & Classes in New York City, GMAT Courses & Classes in Miami, ISEE Courses & Classes in Houston, MCAT Courses & Classes in Washington DC, GRE Courses & Classes in New York City, SSAT Courses & Classes in Los Angeles, GMAT Courses & Classes in New York City, GRE Courses & Classes in Seattle, SAT Courses & Classes in Seattle, ACT Courses & Classes in Phoenix

Popular Test Prep

GRE Test Prep in Houston, GMAT Test Prep in Denver, GRE Test Prep in New York City, ISEE Test Prep in Seattle, ACT Test Prep in Miami, MCAT Test Prep in Chicago, SAT Test Prep in Denver, SAT Test Prep in Philadelphia, GRE Test Prep in Dallas Fort Worth, GMAT Test Prep in San Diego

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 #181 : Derivative Review

Example Question #181 : Derivative Review

Example Question #101 : Derivative At A Point

Example Question #182 : Derivative Review

Example Question #183 : Derivative Review

Example Question #1311 : Calculus Ii

Example Question #181 : Derivative Review

Example Question #186 : Derivative Review

Example Question #187 : Derivative Review

Example Question #182 : Derivative Review

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: