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 #151 : First And Second Derivatives Of Functions

Find the first derivative of the following equation: $y=x^{sin^{2}x}$

Possible Answers:

$\frac{\mathrm{dy} }{\mathrm{d} x}=x^{sinx}(2sinxcosxlnx+\frac{sin^{2}x}{x})$

$\frac{\mathrm{dy} }{\mathrm{d} x}=x^{sin^{2}x}(2sinxcosxlnx+\frac{sin^{2}x}{x})$

$\frac{\mathrm{dy} }{\mathrm{d} x}=x(2sinxcosxlnx+\frac{sin^{2}x}{x})$

$\frac{\mathrm{dy} }{\mathrm{d} x}=x^{xsinx}(2sinxcosxlnx+\frac{sin^{2}x}{x})$

Correct answer:

$\frac{\mathrm{dy} }{\mathrm{d} x}=x^{sin^{2}x}(2sinxcosxlnx+\frac{sin^{2}x}{x})$

Explanation:

Take natural log of both sides: $lny=ln(x^{sin^{2}x})$

Using rules of logs and exponents, bring down $sin^{2}x$ , on right side of equation: $lny=sin^{2}xln(x)$

Differentiate each side: $\frac{\mathrm{d} }{\mathrm{d} x}lny=\frac{\mathrm{d} }{\mathrm{d} x}(sin^{2}xlnx)$

Use implicit differentiation on the left side of the equation, and the product rule on the right side: $\frac{1}{y}\frac{dy}{dx}=\frac{\mathrm{d} }{\mathrm{d} x}(sin^{2})lnx+sin^{2}x\frac{\mathrm{d} }{\mathrm{d} x}lnx$

Simplify: $\frac{1}{y}\frac{dy}{dx}=2sinxcosxlnx+\frac{sin^2{x}}{x}$

Multiply both sides of the equation by : $\frac{dy}{dx}=y(2sinxcosxlnx+\frac{sin^2{x}}{x})$

Simplify (remember that $y=x^{sin^{2}x}$ , from our original equation: $\frac{dy}{dx}=x^{sin^{2}x}(2sinxcosxlnx+\frac{sin^2{x}}{x})$

Report an Error

Example Question #352 : Derivatives

What is the first derivative of f(x)=2x^2-7 ?

Possible Answers:

Correct answer:

Explanation:

Step 1: Use the power rule to simplify the exponent:

2x^2 becomes

Step 2: Take the derivative of the constant:

becomes because the derivative of any constant is zero.

The first derivative is the sum of everything that we have left after taking the derivative:

Final answer is

Report an Error

Example Question #1481 : Calculus Ii

Find the first derivative of: $f(x)=3x^2+4x^{-1}-11$

Possible Answers:

$-6x^{-1}-4x^{2}$

$6x-4x^{-2}$

$6x+4x^{-2}$

$-6x-4x^{-2}$

Correct answer:

$6x-4x^{-2}$

Explanation:

Step 1: Use power rule on the first term:

3x^2 becomes .

Step 2: Use power rule on the second term:

$4x^{-1}$ becomes $-4x^{-2}$

Step 3: Take derivative of the third term:

becomes .

Step 4: Take answers from Steps 1-3...

Final derivative: $6x-4x^{-2}$

Report an Error

Example Question #1483 : Calculus Ii

What is the first derivative of 2x-x^3 ?

Possible Answers:

2-3x^2

-2+3x^2

2+3x^2

-2-3x^2

Correct answer:

2-3x^2

Explanation:

Step 1: Use the power rule on the first term:

becomes

Step 2: Use the power rule on the second term:

-x^3 becomes -3x^2

Step 3: Add the final results for step 1 and step 2:

2-3x^2

Report an Error

Example Question #1482 : Calculus Ii

What is the second derivative of 3x+2x^2-4x^7 ?

Possible Answers:

4-168x^5

-4-128x^5

4+128x^6

3-4x+28x^6

Correct answer:

4-168x^5

Explanation:

Step 1: Use the power rule and take the derivative...

(3x+2x^2-4x^7)'=3+4x-28x^6

Step 2: Use the power rule and take the derivative of the result...

(3+4x-28x^6)'=4-168x^5

Report an Error

Example Question #1485 : Calculus Ii

Find the first derivative of f(x)=3x^4-17x^7+144x^3-20

Possible Answers:

119x^6+12x^3-432x^2

-119x^6+12x^3+432x^2

119x^6-12x^3+432x^2

-119x^6+12x^3-432x^2

Correct answer:

-119x^6+12x^3+432x^2

Explanation:

Step 1: Take derivative of the first term...

3x^4 becomes 12x^3

Step 2: Take derivative of the second term....
-17x^7 becomes -119x^6
Step 3: Take the derivative of the third term....

144x^3 becomes 432x^2
Step 4: Take the derivative of the fourth term...

becomes
Step 5: Rearrange the resulting terms from highest degree to lowest degree.
The degree of a term is the exponent.
So, from highest to lowest, we have 6,3,2.

The answer is: -119x^6+12x^3-432x^2

Report an Error

Example Question #361 : Derivatives

Compute the first derivative of the function:

f(x)=2x^6+3x^4+7x^2+9

Possible Answers:

8x^5+7x^3+9x

12x^5+12x^3+14x

$3x^{12}+4x^7+6x^9$

6x^4+4x^3+2x^2+9x

12x^7+12x^5+14x^3+9x

Correct answer:

12x^5+12x^3+14x

Explanation:

This function is a polynomial made up of powers of x; the power rule for derivatives is:

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

For our equation:

$\frac{d}{dx}(2x^6+3x^4+7x^2+9)=12x^5+12x^3+14x$

Report an Error

Example Question #1487 : Calculus Ii

Compute the first derivative of the following function:

$f(x)=e^{7x}+3e^{2x}+e^5$

Possible Answers:

$7e^{7x-1}+6e^{2x-1}$

$7e^{6x}+6x^x$

$7e^{7x}+6e^{2x}+5e^5$

$7e^{6x}+6e^x+5e^4$

$7e^{7x}+6e^{2x}$

Correct answer:

$7e^{7x}+6e^{2x}$

Explanation:

The rule for derivatives of exponential is:

$\frac{d}{dx}(e^x)=e^x$

Keeping in mind the chain rule, our derivative is:

$\frac{d}{dx}(e^{7x}+3e^{2x}+e^5)=7e^{7x}+6e^{2x}$

Note that because e⁵ is a constant, its derivative is zero.

Report an Error

Example Question #1488 : Calculus Ii

Compute the derivative of the following function:

$f(x)=\ln (5x^2+3x+7)$

Possible Answers:

$\frac{10x+3}{5x^2+3x+7}$

$\frac{1}{10x+3}$

$\frac{1}{5x^2+3x+7}$

$\frac{5x^2+3x+7}{10x+3}$

(5x^2+3x+7)(10x+3)

Correct answer:

$\frac{10x+3}{5x^2+3x+7}$

Explanation:

The derivative rule for natural log functions is:

$\frac{d}{dx}(\ln x)=\frac{1}{x}$

Keeping in mind the chain rule, the derivative is then:

$\frac{d}{dx}(5x^2+3x+7)=\frac{1}{10x+3}\cdot({5x^2+3x+7})$

Report an Error

Example Question #1489 : Calculus Ii

Compute the first derivative of the following function:

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

Possible Answers:

$\sec^2({e^{\sqrt{x}}})$

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

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

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

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

Correct answer:

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

Explanation:

This derivative requires use of the chain rule:

First, taking the derivative of the tangent:

$\tan(e^{\sqrt{x}})\rightarrow\sec^2(e^{\sqrt{x}})$

Next, the derivative of the power of e on the inside of the parenthesis:

$\tan(e^{\sqrt{x}})\rightarrow\sec^2(e^{\sqrt{x}})\rightarrow \sec^2(e^{\sqrt{x}}) \cdot e^{\sqrt{x}}$

Then, the derivative of the square root of x in the power:

$\tan(e^{\sqrt{x}})\rightarrow\sec^2(e^{\sqrt{x}})\rightarrow \sec^2(e^{\sqrt{x}}) \cdot e^{\sqrt{x}} \rightarrow \sec^2(e^{\sqrt{x}}) \cdot e^{\sqrt{x}} \cdot \frac{1}{2\sqrt{x}}$

Combining the terms into one fraction, this simplifies to:

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

Report an Error

View Calculus 2 Tutors

MrG
Certified Tutor

Long Island University-Brooklyn Campus, Master of Science, Applied Mathematics.

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.

View Calculus 2 Tutors

Satweka
Certified Tutor

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

All Calculus 2 Resources

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

Popular Subjects

GRE Tutors in New York City, Calculus Tutors in Los Angeles, GRE Tutors in San Diego, Computer Science Tutors in Dallas Fort Worth, SAT Tutors in San Francisco-Bay Area, English Tutors in San Diego, ACT Tutors in Miami, Calculus Tutors in New York City, ISEE Tutors in Houston, English Tutors in Boston

Popular Courses & Classes

SSAT Courses & Classes in Chicago, GRE Courses & Classes in Boston, SSAT Courses & Classes in Dallas Fort Worth, SAT Courses & Classes in New York City, ISEE Courses & Classes in Los Angeles, LSAT Courses & Classes in Houston, MCAT Courses & Classes in Seattle, MCAT Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in San Francisco-Bay Area, GRE Courses & Classes in Philadelphia

Popular Test Prep

LSAT Test Prep in New York City, SAT Test Prep in San Francisco-Bay Area, LSAT Test Prep in San Francisco-Bay Area, LSAT Test Prep in Houston, SAT Test Prep in Philadelphia, GRE Test Prep in Los Angeles, GRE Test Prep in Seattle, GMAT Test Prep in Miami, GMAT Test Prep in San Francisco-Bay Area, GRE 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 #151 : First And Second Derivatives Of Functions

Example Question #352 : Derivatives

Example Question #1481 : Calculus Ii

Example Question #1483 : Calculus Ii

Example Question #1482 : Calculus Ii

Example Question #1485 : Calculus Ii

Example Question #361 : Derivatives

Example Question #1487 : Calculus Ii

Example Question #1488 : Calculus Ii

Example Question #1489 : Calculus Ii

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: