Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Calculus II

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 #441 : Derivative Review

$\begin{align*}&\text{Calculate the derivative of the function}\\&-180x^{4}ln(13x)\end{align*}$

Possible Answers:

$80x^{3} - 468x^{3}ln(13x)$

$4x^{2}$

$\frac{1}{x}+ 4x^{3}$

$- 720x^{3}ln(13x) - 180x^{3}$

Correct answer:

$- 720x^{3}ln(13x) - 180x^{3}$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }ln(13x)\\&\text{and the second being: }x^{4}\\&\text{With the product scaled by }-180\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[ln(ax)]=\frac{dx}{x}\\&d[x^a]=ax^{a-1}dx\\&\text{From the above we can continue:}\\&u=ln(13x);du=\frac{1}{x}\\&v=x^{4};dv=4x^{3}\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&- 720x^{3}ln(13x) - 180x^{3}\end{align*}$

Report an Error

Example Question #442 : Derivative Review

$\begin{align*}&\text{Find the derivative of the function }\\&f(x)=-42cos(12x)ln(19x)\end{align*}$

Possible Answers:

$504ln(19x)sin(12x) -\frac{ (42cos(12x))}{x}$

$\frac{(3024cos(12x))}{x}- 67032ln(19x)sin(12x)$

$504sin(12x) -\frac{ 42}{x}$

$\frac{(504sin(12x))}{x}$

Correct answer:

$504ln(19x)sin(12x) -\frac{ (42cos(12x))}{x}$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }ln(19x)\\&\text{and the second being: }cos(12x)\\&\text{With the product scaled by }-42\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[ln(ax)]=\frac{dx}{x}\\&d[cos(ax)]=-asin(ax)dx\\&\text{From the above we can continue:}\\&u=ln(19x);du=\frac{1}{x}\\&v=cos(12x);dv=-12sin(12x)\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&504ln(19x)sin(12x) -\frac{ (42cos(12x))}{x}\end{align*}$

Report an Error

Example Question #443 : Derivative Review

$\begin{align*}&\text{Calculate the derivative of the function}\\&-98sin(4x)tan(14x)\end{align*}$

Possible Answers:

$98tan(14x)^{2} - 56cos(4x) + 98$

$392sin(4x)\cdot (14tan(14x)^{2} + 14) + 5488cos(4x)tan(14x)$

$-4cos(4x)\cdot (1372tan(14x)^{2} + 1372)$

$- 98sin(4x)\cdot (14tan(14x)^{2} + 14) - 392cos(4x)tan(14x)$

Correct answer:

$- 98sin(4x)\cdot (14tan(14x)^{2} + 14) - 392cos(4x)tan(14x)$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }-tan(14x)\\&\text{and the second being: }-sin(4x)\\&\text{With the product scaled by }-98\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[tan(ax)]=\frac{adx}{cos^2(ax)}=(atan^2(ax)+a)dx\\&d[sin(ax)]=acos(ax)dx\\&\text{From the above we can continue:}\\&u=-tan(14x);du=- 14tan(14x)^{2} - 14\\&v=-sin(4x);dv=-4cos(4x)\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&- 98sin(4x)\cdot (14tan(14x)^{2} + 14) - 392cos(4x)tan(14x)\end{align*}$

Report an Error

Example Question #444 : Derivative Review

$\begin{align*}&\text{Find the derivative of the function }\\&f(x)=\frac{(120x^{14})}{4^{(4x)}}\end{align*}$

Possible Answers:

$-\frac{ (6720x^{13})}{4^{(4x)}}-\frac{ (6720x^{14}ln(4))}{4^{(4x)}}$

$-\frac{(6720x^{13}ln(4))}{4^{(4x)}}$

$\frac{(40ln(4))}{4^{(4x)}}- 168x^{13}$

$\frac{(1680x^{13})}{4^{(4x)}}-\frac{ (480x^{14}ln(4))}{4^{(4x)}}$

Correct answer:

$\frac{(1680x^{13})}{4^{(4x)}}-\frac{ (480x^{14}ln(4))}{4^{(4x)}}$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }x^{14}\\&\text{and the second being: }\frac{1}{4^{(4x)}}\\&\text{With the product scaled by }120\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[x^a]=ax^{a-1}dx\\&d[b^{ax}]=ab^{ax}ln(b)sx\\&\text{From the above we can continue:}\\&u=x^{14};du=14x^{13}\\&v=\frac{1}{4^{(4x)}};dv=-\frac{(4ln(4))}{4^{(4x)}}\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&\frac{(1680x^{13})}{4^{(4x)}}-\frac{ (480x^{14}ln(4))}{4^{(4x)}}\end{align*}$

Report an Error

Example Question #445 : Derivative Review

$\begin{align*}&\text{Take the derivative with respect to }x\text{ of the function}\\&y=\frac{(108e^{(-4x)})}{3^{(7x)}}\end{align*}$

Possible Answers:

$-\frac{ (432e^{(-4x)})}{3^{(7x)}}-\frac{ (756e^{(-4x)}ln(3))}{3^{(7x)}}$

$\frac{(3024e^{(-4x)})}{3^{(7x)}}+\frac{ (3024e^{(-4x)}ln(3))}{3^{(7x)}}$

$- 36e^{(-4x)} -\frac{ (84ln(3))}{3^{(7x)}}$

$\frac{(3024e^{(-4x)}ln(3))}{3^{(7x)}}$

Correct answer:

$-\frac{ (432e^{(-4x)})}{3^{(7x)}}-\frac{ (756e^{(-4x)}ln(3))}{3^{(7x)}}$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }e^{(-4x)}\\&\text{and the second being: }\frac{1}{3^{(7x)}}\\&\text{With the product scaled by }108\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[e^{ax}]=ae^{ax}dx\\&d[b^{ax}]=ab^{ax}ln(b)sx\\&\text{From the above we can continue:}\\&u=e^{(-4x)};du=-4e^{(-4x)}\\&v=\frac{1}{3^{(7x)}};dv=-\frac{(7ln(3))}{3^{(7x)}}\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&-\frac{ (432e^{(-4x)})}{3^{(7x)}}-\frac{ (756e^{(-4x)}ln(3))}{3^{(7x)}}\end{align*}$

Report an Error

Example Question #1571 : Calculus Ii

What is the derivative of f(x)=4x^3-9x^2+7x-1 ?

Possible Answers:

$f{}'(x)=12x^2-18x-7$

$f{}'(x)=12x^2-8x+7$

$f{}'(x)=12x^2-18x+7$

$f{}'(x)=12x^2+18x+7$

$f{}'(x)=x^2-18x+7$

Correct answer:

$f{}'(x)=12x^2-18x+7$

Explanation:

Recall that when taking the derivative, multiply the exponent by the coefficient in front of the x term and then also subtract one from the exponent:

f(x)=4x^3-9x^2+7x-1

$f'(x)=3\cdot 4x^{3-1}-2\cdot9x^{2-1}+1\cdot7x^{1-1}$

$f{}'(x)=12x^2-18x+7$

Recall that the derivative of a constant is just zero.

Report an Error

Example Question #1572 : Calculus Ii

What is the second derivative of f(x)=16x^2-5x+9 ?

Possible Answers:

f{}''(x)=3

f{}''(x)=32x-5

f{}''(x)=32

f{}''(x)=32x

f{}''(x)=2x

Correct answer:

f{}''(x)=32

Explanation:

Before you can take the second derivative, you must first take the first derivative. Remember to multiply the exponent by the coefficient in front of the x term and then subtract one from the exponent:

f{}'(x)=32x-5

Now, take the second derivative:

f{}''(x)=32

Report an Error

Example Question #251 : First And Second Derivatives Of Functions

Find the first and second derivative of the function.

$f(x)=e^{5x}$

Possible Answers:

$f'(x)=5e^{5x}$

$f''(x)=25e^{5x}$

$f'(x)=5e^{5x}$

$f'(x)=5xe^{5x}$

$f'(x)=e^{5x}$

$f''(x)=e^{5x}$

$f'(x)=5xe^{5x}$

$f''(x)=25x^2e^{5x}$

Correct answer:

$f'(x)=5e^{5x}$

$f''(x)=25e^{5x}$

Explanation:

To find the first and second derivatives, we use the chain rule.

F'(x)=f'(g(x))g'(x)

where F(x)=f(g(x)) .

For the first derivative we have f(x)=e^x and g(x)=5x and so

$f'(x)=e^{5x}*5=5e^{5x}$

For the second derivative we have f(x)=5e^x and g(x)=5x and so

$f''(x)=5e^{5x}*5=25e^x$

Report an Error

Example Question #1574 : Calculus Ii

Find the first and second derivative of the function.

$f(x)=x+sin\,x$

Possible Answers:

$f'(x)=1+cos\,x$

$f''(x)=-sin\,x$

f'(x)=1

f''(x)=0

f'(x)=0

$f''(x)=-sin\,x$

$f'(x)=cos\,x$

$f''(x)=sin\,x$

Correct answer:

$f'(x)=1+cos\,x$

$f''(x)=-sin\,x$

Explanation:

We find the first derivative by deriving term by term. As such,

$f'(x)=\frac{\mathrm{d} }{\mathrm{d} x}[x]+\frac{\mathrm{d} }{\mathrm{d} x}[sin\,x]=1+cos\,x$

Taking the second derivative

$f''(x)=\frac{\mathrm{d} }{\mathrm{d} x}[1]+\frac{\mathrm{d} }{\mathrm{d} x}[cos\,x]=0-sin\,x=-sin\,x$

Report an Error

Example Question #1575 : Calculus Ii

What is the first derivative of f(x)=4x^3-2x^2+x-1?

Possible Answers:

f{}'(x)=12x+1

f{}'(x)=24x-4

$f{}'(x)=12x^2-4x-1$

$f{}'(x)=12x^2-4x+1$

$f{}'(x)=12x^2-4x$

Correct answer:

$f{}'(x)=12x^2-4x+1$

Explanation:

To find the first derivative, look at each term separately.

For the first term, multiply the exponent by the coefficient ( $4\cdot3$ ) and then subtract one from the exponent to get 12x^2 .

Do the same with the next term: $2\cdot-2$ and then subtract one from the exponent to get -4x .

For the third term, the derivative is just the coefficient (1 in this case).

The derivative of a constant is 0.

Put those all together to get your answer: $f{}'(x)=12x^2-4x+1$ .

Report an Error

View Calculus 2 Tutors

Sumit
Certified Tutor

Calcutta University, Bachelor of Science, Mathematics. Indian Statistical Institute, Master of Science, Mathematics.

View Calculus 2 Tutors

Kurosh
Certified Tutor

University of tehran, Bachelor of Engineering, Electrical Engineering.

View Calculus 2 Tutors

Magdy
Certified Tutor

Cairo University, Egypt, Bachelor of Science, Electrical Engineering. New Mexico State University-Main Campus, Doctor of Phil...

All Calculus 2 Resources

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

Popular Subjects

English Tutors in Houston, Physics Tutors in Miami, SSAT Tutors in Washington DC, LSAT Tutors in New York City, French Tutors in Denver, ISEE Tutors in Washington DC, Algebra Tutors in Philadelphia, Math Tutors in San Francisco-Bay Area, SAT Tutors in New York City, Statistics Tutors in Philadelphia

Popular Courses & Classes

Spanish Courses & Classes in New York City, LSAT Courses & Classes in New York City, LSAT Courses & Classes in Chicago, MCAT Courses & Classes in Atlanta, Spanish Courses & Classes in Denver, GRE Courses & Classes in Los Angeles, ISEE Courses & Classes in Phoenix, SSAT Courses & Classes in Chicago, SAT Courses & Classes in Boston, ACT Courses & Classes in Phoenix

Popular Test Prep

SAT Test Prep in Phoenix, LSAT Test Prep in Philadelphia, SSAT Test Prep in Atlanta, ISEE Test Prep in Chicago, MCAT Test Prep in Boston, GRE Test Prep in Philadelphia, GRE Test Prep in San Diego, SAT Test Prep in Philadelphia, ISEE Test Prep in Phoenix, SSAT Test Prep in Boston

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 : Calculus II

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #441 : Derivative Review

Example Question #442 : Derivative Review

Example Question #443 : Derivative Review

Example Question #444 : Derivative Review

Example Question #445 : Derivative Review

Example Question #1571 : Calculus Ii

Example Question #1572 : Calculus Ii

Example Question #251 : First And Second Derivatives Of Functions

Example Question #1574 : Calculus Ii

Example Question #1575 : Calculus Ii

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: