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 #201 : First And Second Derivatives Of Functions

$\begin{align*}&\text{Calculate the derivative of the function}\\&17cos(16sin(6x))\end{align*}$

Possible Answers:

-17sin(16sin(6x))

17cos(96cos(6x))

96cos(6x)

-1632sin(16sin(6x))cos(6x)

Correct answer:

-1632sin(16sin(6x))cos(6x)

Explanation:

$\begin{align*}&\text{Our function, }17cos(16sin(6x))\text{, is one function nested within another.}\\&\text{Note the }16sin(6x)\text{ inside of the }17cos()\\&\text{We'll wish to use the chain rule:}\\&(f\circ g )'=(f'\circ g )\cdot g'\\&\text{Along with the following:}\\&d[sin(u)]=cos(u)du\\&d[cos(u)]=-sin(u)du\\&u=16sin(6x)\\&du=96cos(6x)\\&f'(x)=(96cos(6x))(-17sin(16sin(6x)))=-1632sin(16sin(6x))cos(6x)\end{align*}$

Report an Error

Example Question #201 : First And Second Derivatives Of Functions

$\begin{align*}&\text{Calculate the derivative of the function}\\&-17sin(15sin(x))\end{align*}$

Possible Answers:

-255cos(15sin(x))cos(x)

-17cos(15sin(x))

-17sin(15cos(x))

15cos(x)

Correct answer:

-255cos(15sin(x))cos(x)

Explanation:

$\begin{align*}&\text{Our function, }-17sin(15sin(x))\text{, is one function nested within another.}\\&\text{Note the }15sin(x)\text{ inside of the }-17sin()\\&\text{We'll wish to use the chain rule:}\\&(f\circ g )'=(f'\circ g )\cdot g'\\&\text{Along with the following:}\\&d[sin(u)]=cos(u)du\\&u=15sin(x)\\&du=15cos(x)\\&f'(x)=(15cos(x))(-17cos(15sin(x)))=-255cos(15sin(x))cos(x)\end{align*}$

Report an Error

Example Question #1524 : Calculus Ii

$\begin{align*}&\text{Find the derivative of the function }\\&f(x)=20e^{(5tan(4x))}\end{align*}$

Possible Answers:

$20tan(4x)^{2} + 20$

$20e^{(5tan(4x))}$

$20e^{(5tan(4x))}\cdot (20tan(4x)^{2} + 20)$

$20e^{(20tan(4x)^{2} + 20)}$

Correct answer:

$20e^{(5tan(4x))}\cdot (20tan(4x)^{2} + 20)$

Explanation:

$\begin{align*}&\text{The function }f(x)=20e^{(5tan(4x))}\text{, can be seen as a function nested within another.}\\&\text{Notice the }5tan(4x)\text{ within the }20e^{()}\\&\text{We'll wish to use the chain rule:}\\&(f\circ g )'=(f'\circ g )\cdot g'\\&\text{Along with the following:}\\&d[tan(u)]=\frac{du}{cos^2(u)}=(tan^2(u)+1)du\\&d[e^u]=e^udu\\&u=5tan(4x)\\&du=20tan(4x)^{2} + 20\\&f'(x)=(20tan(4x)^{2} + 20)(20e^{(5tan(4x))})=20e^{(5tan(4x))}\cdot (20tan(4x)^{2} + 20)\end{align*}$

Report an Error

Example Question #402 : Derivative Review

$\begin{align*}&\text{Find the derivative of the function }\\&f(x)=19tan(17cos(5x))\end{align*}$

Possible Answers:

-19tan(85sin(5x))

$-1615sin(5x)\cdot (tan(17cos(5x))^{2} + 1)$

$19tan(17cos(5x))^{2} + 19$

-85sin(5x)

Correct answer:

$-1615sin(5x)\cdot (tan(17cos(5x))^{2} + 1)$

Explanation:

$\begin{align*}&\text{The function }f(x)=19tan(17cos(5x))\text{, can be seen as a function nested within another.}\\&\text{Notice the }17cos(5x)\text{ within the }19tan()\\&\text{We'll wish to use the chain rule:}\\&(f\circ g )'=(f'\circ g )\cdot g'\\&\text{Along with the following:}\\&d[cos(u)]=-sin(u)du\\&d[tan(u)]=\frac{du}{cos^2(u)}=(tan^2(u)+1)du\\&u=17cos(5x)\\&du=-85sin(5x)\\&f'(x)=(-85sin(5x))(19tan(17cos(5x))^{2} + 19)=-1615sin(5x)\cdot (tan(17cos(5x))^{2} + 1)\end{align*}$

Report an Error

Example Question #202 : First And Second Derivatives Of Functions

$\begin{align*}&\text{Calculate the derivative of the function}\\&-6sin(4cos(3x))\end{align*}$

Possible Answers:

6cos(4cos(3x))

72sin(3x)cos(4cos(3x))

6sin(12sin(3x))

Correct answer:

72sin(3x)cos(4cos(3x))

Explanation:

$\begin{align*}&\text{Our function, }-6sin(4cos(3x))\text{, is one function nested within another.}\\&\text{Note the }-4cos(3x)\text{ inside of the }6sin()\\&\text{We'll wish to use the chain rule:}\\&(f\circ g )'=(f'\circ g )\cdot g'\\&\text{Along with the following:}\\&d[cos(u)]=-sin(u)du\\&d[sin(u)]=cos(u)du\\&u=-4cos(3x)\\&du=12sin(3x)\\&f'(x)=(12sin(3x))(6cos(4cos(3x)))=72sin(3x)cos(4cos(3x))\end{align*}$

Report an Error

Example Question #404 : Derivative Review

$\begin{align*}&\text{Take the derivative with respect to }x\text{ of the function}\\&y=-cos(10tan(5x))\end{align*}$

Possible Answers:

$50tan(5x)^{2} + 50$

$-cos(50tan(5x)^{2} + 50)$

$sin(10tan(5x))\cdot (50tan(5x)^{2} + 50)$

sin(10tan(5x))

Correct answer:

$sin(10tan(5x))\cdot (50tan(5x)^{2} + 50)$

Explanation:

$\begin{align*}&\text{The function }y=-cos(10tan(5x))\text{ consists of two functions, one nested in the other..}\\&\text{In particular, }10tan(5x)\text{ is inside of the }-1cos()\\&\text{We'll wish to use the chain rule:}\\&(f\circ g )'=(f'\circ g )\cdot g'\\&\text{Along with the following:}\\&d[tan(u)]=\frac{du}{cos^2(u)}=(tan^2(u)+1)du\\&d[cos(u)]=-sin(u)du\\&u=10tan(5x)\\&du=50tan(5x)^{2} + 50\\&\frac{dy}{dx}=(50tan(5x)^{2} + 50)(sin(10tan(5x)))=sin(10tan(5x))\cdot (50tan(5x)^{2} + 50)\end{align*}$

Report an Error

Example Question #204 : First And Second Derivatives Of Functions

$\begin{align*}&\text{Take the derivative with respect to }x\text{ of the function}\\&y=4cos(7sin(x))\end{align*}$

Possible Answers:

-28sin(7sin(x))cos(x)

-7cos(x)

4sin(7sin(x))

4cos(7cos(x))

Correct answer:

-28sin(7sin(x))cos(x)

Explanation:

$\begin{align*}&\text{The function }y=4cos(7sin(x))\text{ consists of two functions, one nested in the other..}\\&\text{In particular, }-7sin(x)\text{ is inside of the }4cos()\\&\text{We'll wish to use the chain rule:}\\&(f\circ g )'=(f'\circ g )\cdot g'\\&\text{Along with the following:}\\&d[sin(u)]=cos(u)du\\&d[cos(u)]=-sin(u)du\\&u=-7sin(x)\\&du=-7cos(x)\\&\frac{dy}{dx}=(-7cos(x))(4sin(7sin(x)))=-28sin(7sin(x))cos(x)\end{align*}$

Report an Error

Example Question #406 : Derivative Review

$\begin{align*}&\text{Calculate the derivative of the function}\\&10cos(2\cdot 3^x)\end{align*}$

Possible Answers:

$-20\cdot 3^xln(3)sin(2\cdot 3^x)$

$-2\cdot 3^xln(3)$

$10sin(2\cdot 3^x)$

$10cos(2\cdot 3^xln(3))$

Correct answer:

$-20\cdot 3^xln(3)sin(2\cdot 3^x)$

Explanation:

$\begin{align*}&\text{Our function, }10cos(2\cdot 3^x)\text{, is one function nested within another.}\\&\text{Note the }-2\cdot 3^x\text{ inside of the }10cos()\\&\text{We'll wish to use the chain rule:}\\&(f\circ g )'=(f'\circ g )\cdot g'\\&\text{Along with the following:}\\&d[a^u]=a^uduln(a)\\&d[cos(u)]=-sin(u)du\\&u=-2\cdot 3^x\\&du=-2\cdot 3^xln(3)\\&f'(x)=(-2\cdot 3^xln(3))(10sin(2\cdot 3^x))=-20\cdot 3^xln(3)sin(2\cdot 3^x)\end{align*}$

Report an Error

Example Question #407 : Derivative Review

$\begin{align*}&\text{Take the derivative with respect to }x\text{ of the function}\\&y=5sin(3e^{(6x)})\end{align*}$

Possible Answers:

$5sin(18e^{(6x)})$

$90e^{(6x)}cos(3e^{(6x)})$

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

$18e^{(6x)}$

Correct answer:

$90e^{(6x)}cos(3e^{(6x)})$

Explanation:

$\begin{align*}&\text{The function }y=5sin(3e^{(6x)})\text{ consists of two functions, one nested in the other..}\\&\text{In particular, }3e^{(6x)}\text{ is inside of the }5sin()\\&\text{We'll wish to use the chain rule:}\\&(f\circ g )'=(f'\circ g )\cdot g'\\&\text{Along with the following:}\\&d[e^u]=e^udu\\&d[sin(u)]=cos(u)du\\&u=3e^{(6x)}\\&du=18e^{(6x)}\\&\frac{dy}{dx}=(18e^{(6x)})(5cos(3e^{(6x)}))=90e^{(6x)}cos(3e^{(6x)})\end{align*}$

Report an Error

Example Question #403 : Derivatives

The Laplace Transform is an integral transform that converts functions from the time domain to the complex frequency domain . The transformation of a function f(t) into its Laplace Transform F(s) is given by:

$F(s)= \int_{0}^{\infty}e^{-st}*f(t) dt$

Where s=a+bi , where and are constants and is the imaginary number.

Determine the value of $\frac{\mathrm{d} }{\mathrm{d} s}F(s)$ .

Possible Answers:

-tF(s)

-sF(s)

tF(s)

f'(t)*F(s)

Correct answer:

-tF(s)

Explanation:

$\frac{\mathrm{d} }{\mathrm{d} s}F(s)= \frac{d}{ds}\int_{0}^{\infty}e^{-st}*f(t) dt=-t*[\int_{0}^{\infty}e^{-st}*f(t) dt]=-tF(s)$

Report an Error

View Calculus 2 Tutors

Kurosh
Certified Tutor

University of tehran, Bachelor of Engineering, Electrical Engineering.

View Calculus 2 Tutors

James
Certified Tutor

Purdue University-Main Campus, Doctor of Education, Mathematics Teacher Education.

View Calculus 2 Tutors

Jeremy
Certified Tutor

University of Illinois at Chicago, Bachelor of Science, Physics.

All Calculus 2 Resources

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

Popular Subjects

LSAT Tutors in Phoenix, GRE Tutors in Houston, Calculus Tutors in Phoenix, ISEE Tutors in Philadelphia, GRE Tutors in New York City, Computer Science Tutors in Philadelphia, Reading Tutors in Washington DC, LSAT Tutors in San Francisco-Bay Area, Calculus Tutors in Washington DC, Statistics Tutors in San Diego

Popular Courses & Classes

ACT Courses & Classes in Boston, GMAT Courses & Classes in Washington DC, SSAT Courses & Classes in Philadelphia, SAT Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Los Angeles, SSAT Courses & Classes in San Diego, ACT Courses & Classes in Denver, MCAT Courses & Classes in New York City, LSAT Courses & Classes in Denver, SSAT Courses & Classes in Denver

Popular Test Prep

SAT Test Prep in Philadelphia, LSAT Test Prep in Seattle, ISEE Test Prep in San Diego, SAT Test Prep in Dallas Fort Worth, MCAT Test Prep in Dallas Fort Worth, SAT Test Prep in Houston, MCAT Test Prep in Atlanta, ACT Test Prep in Philadelphia, GRE Test Prep in San Francisco-Bay Area, 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 #201 : First And Second Derivatives Of Functions

Example Question #201 : First And Second Derivatives Of Functions

Example Question #1524 : Calculus Ii

Example Question #402 : Derivative Review

Example Question #202 : First And Second Derivatives Of Functions

Example Question #404 : Derivative Review

Example Question #204 : First And Second Derivatives Of Functions

Example Question #406 : Derivative Review

Example Question #407 : Derivative Review

Example Question #403 : Derivatives

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: