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

← Previous 1 2 3 4 5 6 7 8 9 … 25 26 Next →

Example Question #217 : Calculus 3

Find the derivative of the following function:

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

Possible Answers:

$3\sec(3x)\tan(3x)+e^{2x^2+1}$

$3\sec(3x)\tan(3x)-e^{2x^2+1}$

$3\sec(x)\tan(x)+4xe^{2x^2+1}$

$\sec(3x)\tan(3x)+4xe^{2x^2+1}$

$3\sec(3x)\tan(3x)+4xe^{2x^2+1}$

Correct answer:

$3\sec(3x)\tan(3x)+4xe^{2x^2+1}$

Explanation:

The derivative of the function is equal to

$f'(x)=3\sec(3x)\tan(3x)+4xe^{2x^2+1}$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u\frac{\mathrm{du} }{\mathrm{d} x}$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \sec(x)=\sec(x)\tan(x)$

Report an Error

Example Question #1 : Velocity, Speed, Acceleration

Let

$f(x)=e^{\pi x}$

Find the first and second derivative of the function.

Possible Answers:

$f'(x)=\pi e^{\pi x }$

$f''(x)=\pi^2 e^{\pi x}$

$f'(x)=e^{\pi }$

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

$f'(x)=\pi$

$f''(x)=\0$

$f'(x)=e^{\pi x }$

$f''(x)=e^{\pi x}$

Correct answer:

$f'(x)=\pi e^{\pi x }$

$f''(x)=\pi^2 e^{\pi x}$

Explanation:

In order to solve for the first and second derivative, we must use the chain rule.

The chain rule states that if

y=f(u)

and

u=g(x)

then the derivative is

$\frac{dy}{dx}=\frac{dy}{du}*\frac{du}{dx}$

In order to find the first derviative of the function

$f(x)=e^{\pi x}$

we set

y=e^u

and

$u=\pi x$

Because the derivative of the exponential function is the exponential function itself, we get

$\frac{dy}{du}=\frac{d}{du}[e^u]$

$\frac{dy}{du}=e^u$

And differentiating we use the power rule which states

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

As such

$\frac{du}{dx}=\frac{\mathrm{d} }{\mathrm{d} x}[\pi x]$

$\frac{du}{dx}=\pi x^{1-1}=\pi$

And so

$\frac{dy}{dx}=e^{\pi x}*\pi=\pi e^{\pi x}$

$f'(x)=\pi e^{\pi x}$

To solve for the second derivative we set

$y=\pi e^{u}$

and

$u=\pi x$

Because the derivative of the exponential function is the exponential function itself, we get

$\frac{dy}{du}=\frac{d}{du}[\pi e^u]$

$\frac{dy}{du}=\pi e^u$

And differentiating we use the power rule which states

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

As such

$\frac{du}{dx}=\frac{\mathrm{d} }{\mathrm{d} x}[\pi x]$

$\frac{du}{dx}=\pi x^{1-1}=\pi$

And so the second derivative becomes

$\frac{dy}{dx}=\pi e^{\pi x}*\pi=\pi^2 e^{\pi x}$

$f''(x)=\pi^2e^{\pi x}$

Report an Error

Example Question #11 : First And Second Derivatives Of Functions

Find the derivative of the following function:

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

Possible Answers:

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

$-3e^{3x}\sin(e^{3x})+2\sec^2(2x)$

$3e^{3x}\sin(e^{3x})-\sec^2(2x)$

$-\sin(e^{3x})+\sec^2(2x)$

Correct answer:

$-3e^{3x}\sin(e^{3x})+2\sec^2(2x)$

Explanation:

The derivative of the function is equal to

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

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \tan(x)=\sec^2(x)$

Report an Error

Example Question #12 : First And Second Derivatives Of Functions

Find the derivative of the function:

$f(x)=\frac{1}{3x\sqrt{x}}+\sec(x)$

Possible Answers:

$\frac{-6}{\sqrt{x}}-\ln\left |\sec(x)+\tan(x) \right |+C$

$\frac{-3}{2\sqrt{x}}+\sec(x)\tan(x)$

$\frac{-6}{\sqrt{x}}+\ln\left |\sec(x)\tan(x) \right |+C$

$-\frac{6}{\sqrt{x}}+\sec(x)\tan(x)+C$

$\frac{-6}{\sqrt{x}}+\ln\left |\sec(x)+\tan(x) \right |+C$

Correct answer:

$\frac{-6}{\sqrt{x}}+\ln\left |\sec(x)+\tan(x) \right |+C$

Explanation:

The integral of the function is equal to

$\frac{-6}{\sqrt{x}}+\ln\left |\sec(x)+\tan(x) \right |+C$

and was found using the following rules:

$\int x^ndx=\frac{x^{n+1}}{n+1}+C$ , $\int \sec(x)dx=\ln\left | \sec(x)+\tan(x)\right |+C$

Note that it is easier to integrate the first term once it is rewritten as $3x^{-\frac{3}{2}}$ .

Report an Error

Example Question #2 : Velocity, Speed, Acceleration

Find the velocity function of the particle if its position is given by the following function:

$s(t)=e^t\tan(t)+t^5+3$

Possible Answers:

$e^t\tan(t)+e^t\sec^2(t)+t^5$

$e^t\tan(t)+e^t\sec^2(t)+5t^4$

$e^t\tan(t)-e^t\sec^2(t)+5t^4$

$te^t\tan(t)+e^t\sec^2(t)+5t^4$

Correct answer:

$e^t\tan(t)+e^t\sec^2(t)+5t^4$

Explanation:

The velocity function is given by the first derivative of the position function:

$v(t)=s'(t)=e^t\tan(t)+e^t\sec^2(t)+5t^4$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \tan(t)=\sec^2(t)$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^x=e^x$

Report an Error

Example Question #13 : First And Second Derivatives Of Functions

Find the second derivative of the following function:

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

Possible Answers:

$\cos(x)(6x-x^3)-3x^2\sin(x)+4e^{2x}\:$

$\cos(x)(6x-x^3)-6x^2\sin(x)+4e^{2x}$

$\cos(x)(6x+x^3)-6x^2\sin(x)+4e^{2x}$

$\cos(x)(6x-x^3)+6x^2\sin(x)+4e^{2x}$

$\cos(x)(6x-x^3)-6x^2\sin(x)+2e^{2x}$

Correct answer:

$\cos(x)(6x-x^3)-6x^2\sin(x)+4e^{2x}$

Explanation:

The first derivative of the function is equal to

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

The second derivative - the derivative of the function above - of the original function is equal to

$f''(x)=\cos(x)(6x-x^3)-6x^2\sin(x)+4e^{2x}$

Both derivatives were found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u \frac{\mathrm{du} }{\mathrm{d} x}$

Report an Error

Example Question #14 : First And Second Derivatives Of Functions

Find the derivative of the following function:

$f(x)=\sqrt[3]{\cos^2(x)+e^{4x}}$

Possible Answers:

$\frac{-2\cos(x)\sin(x)+4e^{4x}}{3(\cos^2(x)+e^{4x})^{\frac{1}{3}}}$

$\frac{2\cos(x)\sin(x)+4e^{4x}}{3(\cos^2(x)+e^{4x})^{\frac{2}{3}}}$

$\frac{-2\cos(x)\sin(x)+4e^{4x}}{3(\cos^2(x)+e^{4x})^{\frac{2}{3}}}$

$\frac{-\cos(x)\sin(x)+4e^{4x}}{(\cos^2(x)+e^{4x})^{\frac{2}{3}}}$

$\frac{-2\cos(x)\sin(x)+e^{4x}}{3(\cos(x)+e^{4x})^{\frac{2}{3}}}$

Correct answer:

$\frac{-2\cos(x)\sin(x)+4e^{4x}}{3(\cos^2(x)+e^{4x})^{\frac{2}{3}}}$

Explanation:

The derivative was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u\frac{\mathrm{du} }{\mathrm{d} x}$

$f(x)=\sqrt[3]{\cos^2(x)+e^{4x}}$

$f'(x)=\frac{1}{3}(\cos^2(x)+e^{4x})^{-\frac{2}{3}}(-2\cos(x)\sin(x)+4e^{4x})$

$f'(x)=\frac{-2\cos(x)\sin(x)+4e^{4x}}{3(\cos^2(x)+e^{4x})^{\frac{2}{3}}}$

Report an Error

Example Question #15 : First And Second Derivatives Of Functions

Find the derivative of the function:

$f(x)=\frac{e^{2x}+\cos(5x^2)}{e^x}$

Possible Answers:

$\frac{e^{2x}-10x\sin(5x^2)+\cos(5x^2)}{e^x}$

$\frac{e^{3x}-10x\sin(5x^2)+\cos(5x^2)}{e^x}$

$\frac{e^{2x}+10x\sin(5x^2)+\cos(5x^2)}{e^x}$

$\frac{e^{2x}-10xe^x\sin(5x^2)+\cos(5x^2)}{e^x}$

Correct answer:

$\frac{e^{2x}-10x\sin(5x^2)+\cos(5x^2)}{e^x}$

Explanation:

The derivative of the function is equal to

$f(x)=\frac{e^{2x}+\cos(5x^2)}{e^x}$

$f'(x)=\frac{(2e^{2x}-10x\sin(5x^2))e^x-e^x(e^{2x}+\cos(5x^2))}{e^{2x}}$

$f'(x)=\frac{e^{2x}-10x\sin(5x^2)+\cos(5x^2)}{e^x}$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} \frac{f(x)}{g(x)}=\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u \frac{\mathrm{du} }{\mathrm{d} x}$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$

Report an Error

Example Question #16 : First And Second Derivatives Of Functions

Find the derivative of the following function:

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

Possible Answers:

$\frac{e^{\cos(x)}}{2\sqrt{e^\cos(x)}}$

$\frac{-\sin(x)e^{\cos(x)}}{2\sqrt{e^\cos(x)}}$

$\frac{\sin(x)e^{\cos(x)}}{\sqrt{e^\cos(x)}}$

$\frac{-\sin(x)e^{\cos(x)}}{\sqrt{e^\cos(x)}}$

Correct answer:

$\frac{-\sin(x)e^{\cos(x)}}{2\sqrt{e^\cos(x)}}$

Explanation:

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

$f'(x)=\frac{1}{2}(e^{\cos(x)})^{-\frac{1}{2}}(-\sin(x)e^{\cos(x)})$

$f'(x)=\frac{-\sin(x)e^{\cos(x)}}{2\sqrt{e^\cos(x)}}$

The derivative of the function was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u\frac{\mathrm{du} }{\mathrm{d} x}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$

Report an Error

Example Question #1341 : Calculus Ii

Solve for the derivative: y=2cot(2x)

Possible Answers:

$-4sec^2({2x})$

$-4csc^2({2x})$

-4sec({2x})tan({2x})

-4csc({2x})

$-2sec^2({2x})$

Correct answer:

$-4csc^2({2x})$

Explanation:

Write the derivative of cotangent .

cot(x) = -csc^2(x)

The problem requires chain rule since the inner function is . The chain rule is the derivative of the inner function, and the derivative of is .

Take the derivative to obtain $\frac{dy}{dx}$ . Multiply the value obtained by the use of chain rule.

$\frac{dy}{dx} = 2 (-csc^2({2x})) \cdot 2 = -4csc^2({2x})$

Report an Error

← Previous 1 2 3 4 5 6 7 8 9 … 25 26 Next →

View Calculus 2 Tutors

Yoonsik
Certified Tutor

Seoul National University, Bachelor of Science, Physics. University of Pennsylvania, Doctor of Philosophy, Atomic and Molecul...

View Calculus 2 Tutors

Kurosh
Certified Tutor

University of tehran, Bachelor of Engineering, Electrical Engineering.

View Calculus 2 Tutors

Filiberto
Certified Tutor

University of Arizona, Bachelor of Science, Electrical Engineering.

All Calculus 2 Resources

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

Popular Subjects

GMAT Tutors in Houston, SAT Tutors in Boston, Chemistry Tutors in Houston, ACT Tutors in Washington DC, SAT Tutors in San Francisco-Bay Area, ACT Tutors in Miami, SSAT Tutors in Seattle, Math Tutors in Los Angeles, GRE Tutors in Phoenix, Calculus Tutors in Philadelphia

Popular Courses & Classes

ISEE Courses & Classes in Houston, SAT Courses & Classes in Phoenix, ISEE Courses & Classes in Phoenix, SAT Courses & Classes in Boston, ISEE Courses & Classes in Chicago, MCAT Courses & Classes in Washington DC, Spanish Courses & Classes in Denver, ACT Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in San Diego, GRE Courses & Classes in New York City

Popular Test Prep

LSAT Test Prep in San Francisco-Bay Area, SSAT Test Prep in Philadelphia, LSAT Test Prep in Boston, SSAT Test Prep in Atlanta, ACT Test Prep in Los Angeles, ACT Test Prep in Dallas Fort Worth, SAT Test Prep in Boston, SAT Test Prep in Los Angeles, LSAT Test Prep in Chicago, GMAT Test Prep in Denver

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 #217 : Calculus 3

Example Question #1 : Velocity, Speed, Acceleration

Example Question #11 : First And Second Derivatives Of Functions

Example Question #12 : First And Second Derivatives Of Functions

Example Question #2 : Velocity, Speed, Acceleration

Example Question #13 : First And Second Derivatives Of Functions

Example Question #14 : First And Second Derivatives Of Functions

Example Question #15 : First And Second Derivatives Of Functions

Example Question #16 : First And Second Derivatives Of Functions

Example Question #1341 : Calculus Ii

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: