Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

Create An Account

All Calculus 3 Resources

6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #41 : Derivatives

Calculate f'(5) of $f(x)= \sin \left(\pi x \right )$ .

Possible Answers:

$\pi$

$- \pi$

$\frac{\pi}{2}$

Correct answer:

$- \pi$

Explanation:

By the chain rule, we can show

$\frac{d}{dx} \sin \left(f(x) \right )= cos \left(f(x) \right ) \cdot f'(x)$ , or in this case,

$\frac{d}{dx} \sin \left( \pi x \right )= \cos \left( \pi x \right ) \cdot \pi\Rightarrow \cos \left(5 \pi \right ) \cdot \pi = -\pi$ .

Report an Error

Example Question #42 : Derivatives

Find f'(x) given $f(x)=x^{x^2}$

Possible Answers:

$x^{x^2+1}\left(2 \ln x \right )$

$x^{x^2+1}\left(1+2 \ln x \right )$

$x^{x^2+1}$

$x^{x^2}\left(1+2 \ln x \right )$

Correct answer:

$x^{x^2+1}\left(1+2 \ln x \right )$

Explanation:

This derivative must be found using logarithmic differentiation. Consider the following

$\begin{align*} f(x)&=x^{x^2} \\ \log_x f&= x^2\\ \frac{\ln f}{\ln x}&=x^2 \\ \ln f &= x^2 \ln x \\ \frac{f'}{f}&=x+2x \ln x \\ f'&= \left(x + 2x \ln x \right )x^{x^2}= f'(x) = x^{x^2+1}\left(1+2 \ln x \right ) \end{align*}$

Report an Error

Example Question #43 : Derivatives

Find f'(x) given $f(x)= 7^{3x}$

Possible Answers:

$3 \cdot 7^{2x}$

$3 \cdot 7^{3x}$

$7^{3x} \cdot \ln 7^3$

$\ln 7^3$

Correct answer:

$7^{3x} \cdot \ln 7^3$

Explanation:

This derivative is most easily done by logarithmic differentiation. Consider the following

$\begin{align*} f(x)&=7^{3x} \\ \ln f&=\ln 7 \cdot 3x \\ \frac{f'}{f}&=\ln 7 \cdot 3 \\ f'(x)&=7^{3x} \cdot 3 \ln 7 \\ f'(x)&=7^{3x} \cdot \ln 7^3 \end{align*}$

Report an Error

Example Question #44 : Derivatives

Find the derivative of: y=-cos(5x+2)

Possible Answers:

5sin(5x+2)

5sin(5x+2)cos(5x+2)

-5sin(5x+2)

-sin(5x+2)

-5sin(5x+2)cos(5x+2)

Correct answer:

5sin(5x+2)

Explanation:

Take the derivative of the cosine function. This will also require chain rule, which is the derivative of the inner function.

$\frac{dy}{dx} = -[-sin(5x+2)] \cdot 5$

The answer is: $\frac{dy}{dx}= 5sin(5x+2)$

Report an Error

Example Question #45 : Calculus Review

Find the derivative: ln(cos(2x))

Possible Answers:

tan(2x)

$-\frac{2}{sin(2x)}$

$\frac{1}{sin(2x)}$

-2tan(2x)

$\frac{1}{cos(2x)}$

Correct answer:

-2tan(2x)

Explanation:

This problem will require multiple chain rule. Take the derivative of natural log, and then apply the inner function of the natural log, cosine, and then also apply the chain rule for .

$\frac{d}{dx}ln(cos(2x)) = \frac{1}{cos(2x)} \cdot -sin(2x) \cdot 2$

Combine this into one term and simplify.

$-\frac{2sin(2x)}{cos(2x)} = -2tan(2x)$

The answer is: -2tan(2x)

Report an Error

Example Question #45 : Calculus Review

Evaluate $\frac{d}{dx}(yx+y)$

Possible Answers:

None of the other answers

y+x

y+1

x+1

Correct answer:

None of the other answers

Explanation:

The correct answer is (x+1)y'+y .

Since we are taking the (full) derivative of an expression involving and , we must use implicit differentiation. We proceed as follows

$\frac{d}{dx}(yx+y)$ start.

$=\frac{d}{dx}(yx)+\frac{d}{dx}(y)$

=(xy'+(1)y)+y' . Using $\frac{d}{dx}y=y'$ , and the Product Rule.

=(1+x)y' +y . Factor

Report an Error

Example Question #48 : Calculus Review

Evaluate the limit

$\lim_{h \to 0}\frac{12(\frac{1}{2}+h)^{12}+12(\frac{1}{2})^{12}}{h}$

by interpreting the limit in terms of the definition of the derivative.

Possible Answers:

$\frac{9}{128}$

The limit does not exist.

Correct answer:

$\frac{9}{128}$

Explanation:

The definition of the derivative is

$\lim_{h \to 0}\frac{f(a+h)-f(a)}{h} = f'(a)$ .

Equating

$\lim_{h \to 0}\frac{12(\frac{1}{2}+h)^{12}+12(\frac{1}{2})^{12}}{h}$ , to the definition of the derivative, we have

$\lim_{h \to 0}\frac{12(\frac{1}{2}+h)^{12}+12(\frac{1}{2})^{12}}{h} = \frac{d}{dx}(12x^{12})|_{x=1/2}=144x^{11}|_{x=1/2}$

$=144(\frac{1}{2})^{11}=\frac{144}{2048} =\frac{9}{128}$ .

Report an Error

Example Question #45 : Derivatives

What is the derivative of $5x^4+2x^2+\frac {3}{x}$ ?

Possible Answers:

$20x^3-4x-\frac {3}{x^2}$

$20x^3+4x-\frac {3}{x^2}$

$-20x^3+4x-\frac {3}{x^2}$

$20x^3+4x-\frac {3}{x}$

Correct answer:

$20x^3+4x-\frac {3}{x^2}$

Explanation:

Step 1: Take the derivative of the first term: 5x^4

$(5x^4)'=20x^{4-1}=20x^3$ via the power rule.
Step 2: Take the derivative of the next term: 2x^2
$(2x^2)'=4x^{2-1}=4x$ , again using the product rule.
Step 3: Take the derivative of the last term $\frac {3}{x}$ .

We will have to use Quotient Rule here.

Find the derivative of f(x) and g(x):

f'(x)=0 , g'(x)=1 .

Use the formula: $\frac {f'(x)g(x)-g'(x)f(x)}{[g(x)]^2}$

$\frac {0(x)-1(3)}{(x)^2}=\frac {-3}{x^2}$ .

Step 4: Put all the final terms from highest to lowest degree...

We get: $20x^3+4x-\frac {3}{x^2}$ .

Report an Error

Example Question #45 : Derivatives

Find the derivative of

$\sin^n (x^m)$

Possible Answers:

$m^2n^2(x^{m})\sin^{n}(x^m)$

$n(x^{m-1})\sin^{n-1}(x^m)$

$(x^{m-1})\sin^{n-1}(x^m)$

$mn(x^{m-1})\sin^{n-1}(x^m)$

Correct answer:

$mn(x^{m-1})\sin^{n-1}(x^m)$

Explanation:

Untitled

Report an Error

Example Question #51 : Calculus Review

Find the derivative of the following function:

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

Possible Answers:

$f'(x)=e^{x}\sec x-e^{x}\sec x \tan x+6x$

$f'(x)=xe^{x}\sec x+e^{x}\sec x \tan x+6x$

$f'(x)=e^{x}\sec x+e^{x}\sec x \tan x+6x$

$f'(x)=e^{x}\sec x+e^{x}\tan^2 x+6x$

Correct answer:

$f'(x)=e^{x}\sec x+e^{x}\sec x \tan x+6x$

Explanation:

The derivative can be taken by taking the individual derivatives of each component of the function. The first component requires the use of the product rule, which states that:

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

Following this rule, the derivative of the first component is:

$e^{x}\sec x+e^{x}\sec x\tan x$

The derivatives were found by using the following rules:

$\frac{d}{du}e^{u}=e^{u}\cdot u'$ , $\frac{d}{dx}\sec x=\sec x\tan x$

Then, the derivative of the last component is found by using the power rule which states that:

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

The derivative of the final component is

With each individual component found, they can be added together to get the final answer:

$f'(x)=e^{x}\sec x+e^{x}\sec x \tan x+6x$

Report an Error

View Calculus 3 Tutors

Jozsef
Certified Tutor

University of Waterloo, Doctor of Philosophy, Applied Mathematics.

View Calculus 3 Tutors

Satweka
Certified Tutor

The University of Texas at Dallas, Master of Science, Biomedical Engineering.

View Calculus 3 Tutors

Jack
Certified Tutor

University of Washington-Seattle Campus, Bachelor in Arts, Mathematics. Western Washington University, Master of Science, Mat...

All Calculus 3 Resources

6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

French Tutors in New York City, Calculus Tutors in New York City, Statistics Tutors in Philadelphia, GRE Tutors in New York City, Physics Tutors in New York City, Biology Tutors in Miami, ISEE Tutors in Seattle, Reading Tutors in Houston, SSAT Tutors in Los Angeles, ACT Tutors in Houston

Popular Courses & Classes

GMAT Courses & Classes in Seattle, ISEE Courses & Classes in Atlanta, Spanish Courses & Classes in Seattle, SAT Courses & Classes in San Diego, LSAT Courses & Classes in Houston, LSAT Courses & Classes in Chicago, MCAT Courses & Classes in New York City, SSAT Courses & Classes in Los Angeles, MCAT Courses & Classes in Seattle, SSAT Courses & Classes in Chicago

Popular Test Prep

SAT Test Prep in Boston, LSAT Test Prep in New York City, SSAT Test Prep in San Francisco-Bay Area, GMAT Test Prep in Atlanta, LSAT Test Prep in Denver, ISEE Test Prep in San Diego, LSAT Test Prep in Boston, SAT Test Prep in Houston, ISEE Test Prep in Atlanta, SAT Test Prep in Los Angeles

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 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #41 : Derivatives

Example Question #42 : Derivatives

Example Question #43 : Derivatives

Example Question #44 : Derivatives

Example Question #45 : Calculus Review

Example Question #45 : Calculus Review

Example Question #48 : Calculus Review

Example Question #45 : Derivatives

Example Question #45 : Derivatives

Example Question #51 : Calculus Review

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: