We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus AB : Computation of the Derivative

Study concepts, example questions & explanations for AP Calculus AB

Create An Account

All AP Calculus AB Resources

3 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

← Previous 1 2 … 23 24 25 26 27 28 29 30 31 Next →

Example Question #58 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the following function using the quotient rule:

$f=\frac{e^x}{\tan{x}}$

Possible Answers:

$f'=\frac{e^x\tan{x}-e^x\sec^2{x}}{\tan^2{x}}$

$f'=\frac{e^x\tan{x}-e^x\sec^2{x}}{\tan{x}}$

$f'=\frac{e^x\tan{x}+e^x\sec^2{x}}{\tan^2{x}}$

$f'=\frac{xe^x\tan{x}-e^x\sec^2{x}}{\tan^2{x}}$

Correct answer:

$f'=\frac{e^x\tan{x}-e^x\sec^2{x}}{\tan^2{x}}$

Explanation:

To find the derivative of the function using the quotient rule, we apply the following definition:

$\frac{\mathrm{d} }{\mathrm{d} x}(\frac{f}{g})=\frac{gf'-fg'}{g^2}$

$\frac{\mathrm{d} }{\mathrm{d} x}(\frac{e^x}{\tan{x}})=\frac{\tan{x}*\frac{\mathrm{d} }{\mathrm{d} x}(e^x)-e^x*\frac{\mathrm{d} }{\mathrm{d} x}(\tan{x})}{\tan^2{x}}=\frac{e^x\tan{x}-e^x\sec^2{x}}{\tan^2{x}}$

Report an Error

Example Question #52 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the following function:

f=10x^3+5x^2-1

Possible Answers:

f'=30x^2+10x

f'=3x^2+x

f'=30x^2+10x-1

f'=30x^3+10x^2

Correct answer:

f'=30x^2+10x

Explanation:

To find the derivative of the function, we take the derivative of each element in the function independently, then add them up.

Using $\frac{\mathrm{d} }{\mathrm{d} x}x^n=nx^{n-1}$ , we solve

$\frac{\mathrm{d} }{\mathrm{d} x}(10x^3)+\frac{\mathrm{d} }{\mathrm{d} x}(5x^2)+\frac{\mathrm{d} }{\mathrm{d} x}(-1)=30x^2+10x$

Report an Error

Example Question #51 : Derivative Rules For Sums, Products, And Quotients Of Functions

Determine the second derivative of f(x)=ln(x^2-1)

Possible Answers:

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

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

$f''(x)=\frac{x}{2x -1}$

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

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

Correct answer:

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

Explanation:

Finding our second derivative requires two steps, we first must find the derivative then find the corresponding rate of change for that new equation.

f(x)=ln(x^2-1)

Here, the chain rule is used since our function is of the form [f(g(x))]dx=f'(x)g'(f(x))

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

We now must use the quotient rule since our function is a rational function. We use the rule $[\frac{f(x)}{g(x)}]dx=\frac{f'(x)g(x)-g'(x)f(x))}{(g(x))^2}$

Therefore,

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

Report an Error

Example Question #6 : Finding Derivative Of A Function

What is the derivative of x^3+2x+5 ?

Possible Answers:

3x^2+2x

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

3x^2+2

Correct answer:

3x^2+2

Explanation:

To solve this problem, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.

We're going to treat as 5x^0 , as anything to the zero power is one.

$\frac{\mathrm{d} }{\mathrm{d} x}(x^3+2x+5)=(3*x^{3-1})+(1*2x^{1-1})+(0*5x^{0-1})$

Notice that $(0*5x^{0-1})=0$ , as anything times zero is zero.

$\frac{\mathrm{d} }{\mathrm{d} x}(x^3+2x+5)=(3*x^{3-1})+(1*2x^{1-1})$

$\frac{\mathrm{d} }{\mathrm{d} x}(x^3+2x+5)=(3*x^{2})+(2x^{0})$

$\frac{\mathrm{d} }{\mathrm{d} x}(x^3+2x+5)=3x^2+2$

Report an Error

Example Question #61 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the slope of the $f(x)=\sqrt{x^2 +2x-3}$ at x=2 .

Possible Answers:

$\frac{3\sqrt{5}}{5}$

$5$

$\frac{6\sqrt{5}}{5}$

$6$

$\frac{\sqrt{5}}{5}$

Correct answer:

$\frac{3\sqrt{5}}{5}$

Explanation:

First we need to find the derivative of the function. $f'(x)=\frac{1+x}{\sqrt{-3+2x+x^2}}$

Now, we can plug in x=2 to the derivative function.

$f'(2)=\frac{3\sqrt{5}}{5}$

Report an Error

Example Question #62 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the following function:

$f=\frac{x^4\sin{x}}{x^2+3}$

Possible Answers:

$f'=\frac{(x^2+3)(4x^3\sin(x)+x^4\cos(x)-2x^5\sin(x))}{(x^2-3)^2}$

$f'=\frac{(x^2-3)(4x^3\sin(x)+x^4\cos(x)+2x^5\sin(x))}{(x^2+3)^2}$

$f'=\frac{(x^2+3)(4x^3\sin(x)+x^4\cos(x)-2x^5\cos(x))}{(x^2+3)^2}$

$f'=\frac{(x^2+3)(4x^3\sin(x)+x^4\cos(x)-2x^5\sin(x))}{(x^2+3)^2}$

Correct answer:

$f'=\frac{(x^2+3)(4x^3\sin(x)+x^4\cos(x)-2x^5\sin(x))}{(x^2+3)^2}$

Explanation:

To find the derivative of the function, we use a combination of the quotient rule and product rule

$f=\frac{x^4\sin{x}}{x^2+3}$

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

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

$f'=\frac{(x^2+3)(4x^3\sin(x)+x^4\cos(x)-2x^5\sin(x))}{(x^2+3)^2}$

Report an Error

Example Question #741 : Ap Calculus Ab

Find the derivative of the function

f=x^3+5x^2+8x+20

Possible Answers:

$f'=\frac{3x^2+10x^2+8}{x}$

f'=3x^2+10x+8

f'=3x^3+10x^2+8x

f'=3x^2-10x+8

Correct answer:

f'=3x^2+10x+8

Explanation:

To find the derivative of the function, we use the rule

$\frac{\mathrm{d} }{\mathrm{d} x}x^n=nx^{n-1}$ and apply it to each term in the function

$f'=\frac{\mathrm{d} }{\mathrm{d} x}(x^3)+\frac{\mathrm{d} }{\mathrm{d} x}(5x^2)+\frac{\mathrm{d} }{\mathrm{d} x}(8x)+\frac{\mathrm{d} }{\mathrm{d} x}(20)$

f'=3x^2+10x+8

Report an Error

Example Question #751 : Ap Calculus Ab

Find the derivative of the following function

$f=\frac{x^3}{\tan{x}}$

Possible Answers:

$f'=\frac{3x^2\tan{x}-x^3\sec^2{x}}{\tan{x}}$

$f'=\frac{3x^2\tan{x}+x^3\sec^2{x}}{\tan^2{x}}$

$f'=\frac{3x^2\tan{x}-x^3\sec^2{x}}{\tan^2{x}}$

$f'=\frac{x^3\tan{x}-x^3\sec^2{x}}{\tan^2{x}}$

Correct answer:

$f'=\frac{3x^2\tan{x}-x^3\sec^2{x}}{\tan^2{x}}$

Explanation:

To find the derivative of the function, we use the quotient rule, which is

$\frac{\mathrm{d} }{\mathrm{d} x}(\frac{f}{g})=\frac{gf'-fg'}{g^2}$

Applying this to the function from the problem statement, we get

$f'=\frac{(x^3)'\tan{x}-x^3(\tan{x})'}{\tan^2{x}}$

$f'=\frac{3x^2\tan{x}-x^3\sec^2{x}}{\tan^2{x}}$

Report an Error

Example Question #752 : Ap Calculus Ab

Find the derivative of the function

$f=x\sin{x}\cos{x}$

Hint: $\cos^2{x}-\sin^2{x}=\cos{2x}$

Possible Answers:

$f'=\sin{x}\cos{x}+x\cos{2x}$

$f'=\sin{x}\cos^2{x}+x\cos{2x}$

$f'=\sin{x}\cos{x}+x\cos{x}$

$f'=\sin{x}\cos{x}+\cos{2x}$

Correct answer:

$f'=\sin{x}\cos{x}+x\cos{2x}$

Explanation:

To find the derivative of the function, we use the product rule, which by definition is

$\frac{\mathrm{d} }{\mathrm{d} x}(f*g)=f'g+g'f$

In this case, we can split up the product such that

$f=(x\sin{x})*(\cos{x})$

$f'=(x\sin{x})'(\cos{x})+(x\sin{x})(\cos{x})'$

$y'=(\sin{x+x\cos{x}})(\cos{x})+\sin{x}(x\sin{x})$

$y'=\sin{x\cos{x}}+x\cos^2{x}+x\sin^2{x}$

$f'=\sin{x}\cos{x}+x\cos{2x}$

Report an Error

Example Question #61 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the function:

y=x^3e^x

Possible Answers:

$y'=\frac{e^x}{(x^3+3x^2)}$

$y'=\frac{-e^x}{(x^3+3x^2)}$

y'=-e^x(x^3+3x^2)

y'=e^x(x^3+3x^2)

y'=3x^2e^x

Correct answer:

y'=e^x(x^3+3x^2)

Explanation:

This question requires us to understand two things.

First, the derivative of e^x is always e^x , such that the exponent contains a single variable (any other operation or numerical factors could cause the chain rule to come into play; a later topic)

Second, we must understand the product rule for derivatives. The product rule works as follows:

$\frac{dy}{dx}f(x)*g(x)=f(x)g'(x)+g(x)f'(x)$

Understanding these two concepts allows us to tackle the derivative of the given function.

y=x^3e^x

y'=(x^3)(e^x) +(e^x)(3x^2)

y'=e^x(x^3) +e^x(3x^2)

This simplifies to:

y'=e^x(x^3+3x^2)

We are finished taking the derivative of the product!

Report an Error

← Previous 1 2 … 23 24 25 26 27 28 29 30 31 Next →

View AP Calculus AB Tutors

Nora
Certified Tutor

University of Cincinnati-Main Campus, Bachelor of Technology, Biomedical Sciences.

View AP Calculus AB Tutors

Andrew
Certified Tutor

Louisiana State University Eunice, Bachelor of Science, Computer Software Engineering.

View AP Calculus AB Tutors

Gregory
Certified Tutor

Pennsylvania State University-Main Campus, Bachelor of Science, Chemical Engineering. Carnegie Mellon University, Doctor of P...

All AP Calculus AB Resources

3 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Calculus Tutors in Atlanta, Computer Science Tutors in Boston, Reading Tutors in Houston, ISEE Tutors in Seattle, LSAT Tutors in Miami, GRE Tutors in New York City, ACT Tutors in Los Angeles, Chemistry Tutors in Houston, GMAT Tutors in Atlanta, SAT Tutors in Washington DC

Popular Courses & Classes

ACT Courses & Classes in San Diego, LSAT Courses & Classes in Seattle, Spanish Courses & Classes in Boston, ISEE Courses & Classes in Philadelphia, SAT Courses & Classes in Chicago, GMAT Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in Los Angeles, ISEE Courses & Classes in Chicago, SSAT Courses & Classes in Atlanta, GRE Courses & Classes in San Francisco-Bay Area

Popular Test Prep

SSAT Test Prep in Seattle, ACT Test Prep in San Diego, ACT Test Prep in Miami, ISEE Test Prep in Phoenix, MCAT Test Prep in Philadelphia, SAT Test Prep in Dallas Fort Worth, MCAT Test Prep in Washington DC, SAT Test Prep in New York City, LSAT Test Prep in New York City, GRE Test Prep in San Francisco-Bay Area

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

AP Calculus AB : Computation of the Derivative

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #58 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #52 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #51 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #6 : Finding Derivative Of A Function

Example Question #61 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #62 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #741 : Ap Calculus Ab

Example Question #751 : Ap Calculus Ab

Example Question #752 : Ap Calculus Ab

Example Question #61 : Derivative Rules For Sums, Products, And Quotients Of Functions

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: