Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : First and Second Derivatives of Functions

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

$\begin{align*}&\text{Take the derivative with respect to }x\text{ of the function}\\&y=-70sin(15x)tan(5x)\end{align*}$

Possible Answers:

$5tan(5x)^{2} - 15cos(15x) + 5$

$-15cos(15x)\cdot (5tan(5x)^{2} + 5)$

$- 70sin(15x)\cdot (5tan(5x)^{2} + 5) - 1050cos(15x)tan(5x)$

$525cos(15x)tan(5x) - 150sin(15x)\cdot (5tan(5x)^{2} + 5)$

Correct answer:

$- 70sin(15x)\cdot (5tan(5x)^{2} + 5) - 1050cos(15x)tan(5x)$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }-sin(15x)\\&\text{and the second being: }tan(5x)\\&\text{With the product scaled by }70\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[sin(ax)]=acos(ax)dx\\&d[tan(ax)]=\frac{adx}{cos^2(ax)}=(atan^2(ax)+a)dx\\&\text{From the above we can continue:}\\&u=-sin(15x);du=-15cos(15x)\\&v=tan(5x);dv=5tan(5x)^{2} + 5\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&- 70sin(15x)\cdot (5tan(5x)^{2} + 5) - 1050cos(15x)tan(5x)\end{align*}$

Report an Error

Example Question #441 : Derivatives

$\begin{align*}&\text{Take the derivative with respect to }x\text{ of the function}\\&y=170ln(-13x)tan(5x)\end{align*}$

Possible Answers:

$5tan(5x)^{2} +\frac{ 1}{x}+ 5$

$130ln(-13x)\cdot (5tan(5x)^{2} + 5) -\frac{ (85tan(5x))}{x}$

$\frac{(5tan(5x)^{2} + 5)}{x}$

$170ln(-13x)\cdot (5tan(5x)^{2} + 5) +\frac{ (170tan(5x))}{x}$

Correct answer:

$170ln(-13x)\cdot (5tan(5x)^{2} + 5) +\frac{ (170tan(5x))}{x}$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }ln(-13x)\\&\text{and the second being: }tan(5x)\\&\text{With the product scaled by }170\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[ln(ax)]=\frac{dx}{x}\\&d[tan(ax)]=\frac{adx}{cos^2(ax)}=(atan^2(ax)+a)dx\\&\text{From the above we can continue:}\\&u=ln(-13x);du=\frac{1}{x}\\&v=tan(5x);dv=5tan(5x)^{2} + 5\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&170ln(-13x)\cdot (5tan(5x)^{2} + 5) +\frac{ (170tan(5x))}{x}\end{align*}$

Report an Error

Example Question #442 : Derivative Review

$\begin{align*}&\text{Calculate the derivative of the function}\\&\frac{(100ln(14x))}{x^{20}}\end{align*}$

Possible Answers:

$\frac{100}{x^{21}}-\frac{ (2000ln(14x))}{x^{21}}$

$-\frac{20}{x^{22}}$

$-\frac{ (2800ln(14x))}{x^{21}}-\frac{ 200}{x^{21}}$

$\frac{1}{x}-\frac{ 20}{x^{21}}$

Correct answer:

$\frac{100}{x^{21}}-\frac{ (2000ln(14x))}{x^{21}}$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }\frac{1}{x^{20}}\\&\text{and the second being: }ln(14x)\\&\text{With the product scaled by }100\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[x^a]=ax^{a-1}dx\\&d[ln(ax)]=\frac{dx}{x}\\&\text{From the above we can continue:}\\&u=\frac{1}{x^{20}};du=-\frac{20}{x^{21}}\\&v=ln(14x);dv=\frac{1}{x}\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&\frac{100}{x^{21}}-\frac{ (2000ln(14x))}{x^{21}}\end{align*}$

Report an Error

Example Question #441 : Derivatives

$\begin{align*}&\text{Take the derivative with respect to }x\text{ of the function}\\&y=28ln(-17x)e^{(-20x)}\end{align*}$

Possible Answers:

$\frac{(28e^{(-20x)})}{x}- 560ln(-17x)e^{(-20x)}$

$\frac{(280e^{(-20x)})}{x}- 680ln(-17x)e^{(-20x)}$

$\frac{1}{x}- 20e^{(-20x)}$

$-\frac{(20e^{(-20x)})}{x}$

Correct answer:

$\frac{(28e^{(-20x)})}{x}- 560ln(-17x)e^{(-20x)}$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }ln(-17x)\\&\text{and the second being: }e^{(-20x)}\\&\text{With the product scaled by }28\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[ln(ax)]=\frac{dx}{x}\\&d[e^{ax}]=ae^{ax}dx\\&\text{From the above we can continue:}\\&u=ln(-17x);du=\frac{1}{x}\\&v=e^{(-20x)};dv=-20e^{(-20x)}\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&\frac{(28e^{(-20x)})}{x}- 560ln(-17x)e^{(-20x)}\end{align*}$

Report an Error

Example Question #441 : Derivatives

$\begin{align*}&\text{Find the derivative of the function }\\&f(x)=72ln(3x)e^{(17x)}\end{align*}$

Possible Answers:

$\frac{(72e^{(17x)})}{x}+ 1224ln(3x)e^{(17x)}$

$17e^{(17x)} +\frac{ 1}{x}$

$\frac{(17e^{(17x)})}{x}$

$-\frac{ (136e^{(17x)})}{x}- 459ln(3x)e^{(17x)}$

Correct answer:

$\frac{(72e^{(17x)})}{x}+ 1224ln(3x)e^{(17x)}$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }ln(3x)\\&\text{and the second being: }e^{(17x)}\\&\text{With the product scaled by }72\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[ln(ax)]=\frac{dx}{x}\\&d[e^{ax}]=ae^{ax}dx\\&\text{From the above we can continue:}\\&u=ln(3x);du=\frac{1}{x}\\&v=e^{(17x)};dv=17e^{(17x)}\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&\frac{(72e^{(17x)})}{x}+ 1224ln(3x)e^{(17x)}\end{align*}$

Report an Error

Example Question #443 : Derivative Review

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

Possible Answers:

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

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

$4x^{2}$

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

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 #444 : Derivative Review

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

Possible Answers:

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

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

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

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

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 #445 : Derivative Review

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

Possible Answers:

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

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

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

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

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 #241 : First And Second Derivatives Of Functions

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

Possible Answers:

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

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

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

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

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 #446 : 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:

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

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

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

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

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

View Calculus 2 Tutors

Rodolfo
Certified Tutor

The University of Texas at San Antonio, Bachelor of Science, Mathematics.

View Calculus 2 Tutors

Paul
Certified Tutor

University of Phoenix-Albuquerque Campus, Bachelor of Science, Computer and Information Systems Security.

View Calculus 2 Tutors

Ping
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Computer Science. University of North Texas, Master of Arts, Education.

All Calculus 2 Resources

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

Popular Subjects

Algebra Tutors in San Diego, Calculus Tutors in New York City, SAT Tutors in Phoenix, Reading Tutors in Houston, English Tutors in Chicago, ISEE Tutors in Atlanta, LSAT Tutors in San Francisco-Bay Area, SSAT Tutors in Atlanta, Reading Tutors in Washington DC, Statistics Tutors in Denver

Popular Courses & Classes

GMAT Courses & Classes in New York City, SAT Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in Phoenix, Spanish Courses & Classes in San Francisco-Bay Area, Spanish Courses & Classes in San Diego, MCAT Courses & Classes in Atlanta, SAT Courses & Classes in Philadelphia, GRE Courses & Classes in Washington DC, LSAT Courses & Classes in San Diego, SAT Courses & Classes in Los Angeles

Popular Test Prep

MCAT Test Prep in Phoenix, MCAT Test Prep in Seattle, ACT Test Prep in Washington DC, GMAT Test Prep in Seattle, SAT Test Prep in San Diego, GMAT Test Prep in San Francisco-Bay Area, GRE Test Prep in Denver, ISEE Test Prep in Phoenix, GMAT Test Prep in Denver, LSAT 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 : First and Second Derivatives of Functions

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #441 : Derivative Review

Example Question #441 : Derivatives

Example Question #442 : Derivative Review

Example Question #441 : Derivatives

Example Question #441 : Derivatives

Example Question #443 : Derivative Review

Example Question #444 : Derivative Review

Example Question #445 : Derivative Review

Example Question #241 : First And Second Derivatives Of Functions

Example Question #446 : Derivative Review

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: