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

Example Question #457 : Derivatives

Find the derivative of the function:

$f(x)=\frac{8x^2}{4-x}$

Possible Answers:

$f'(x)=-\frac{4x(4-x)}{(8-x)^2}$

$f'(x)=-\frac{x(8-x)}{(4-x)^3}$

$f'(x)=\frac{4x(4-x)}{(8-x)^2}$

$f'(x)=\frac{8x(8-x)}{(4-x)^2}$

$f'(x)=-\frac{x(4-x)}{(8-x)^3}$

Correct answer:

$f'(x)=\frac{8x(8-x)}{(4-x)^2}$

Explanation:

*****************************************************************

STEPS

*****************************************************************

Given:

$f(x)=\frac{8x^2}{4-x}$

Set the derivative up with the quotient rule:

$f'(\frac{u}{v})=\frac{v(u')-u(v')}{v^2}$

$f'(x)=\frac{(4-x)(16x)-(8x^2)(-1)}{(4-x)^2}$

Multiply factors:

$f'(x)=\frac{64x-16x^2+8x^2}{(4-x)^2}$

Combine like terms:

$f'(x)=\frac{64x-8x^2}{(4-x)^2}$

Factoring out 8x in the numerator, we arrive at the correct answer:

*****************************************************************

ANSWER

*****************************************************************

${\color{Red} f'(x)=\frac{8x(8-x)}{(4-x)^2}}$

Report an Error

Example Question #458 : Derivatives

Find the derivative of the function:

r(s)=s^3-s^2sin(s)

Simplify your final answer through factoring terms out

Possible Answers:

r'(s)=5s(s-cos(s)-2sin(s))

r'(s)=-5s(s+cos(s)+2sin(s))

r'(s)=s(3s-scos(s)-2sin(s))

r'(s)=3s(5s-3cos(s)-2sin(s))

$r'(s)=-\frac{1}{5s}(s+cos(s)+2sin(s))$

Correct answer:

r'(s)=s(3s-scos(s)-2sin(s))

Explanation:

*****************************************************************

STEPS

*****************************************************************

Given:

r(s)=s^3-s^2sin(s)

r'(s)=d(s^3)-d(s^2sin(s))

r'(s)=3s^2-d(s^2sin(s))

Set up a product rule for the derivative of the second term:

$d(s^2sin(s))={\color{DarkBlue} [s^2(cos(s))+2s(sin(s))]}$

Plug this new-found product in for d(s^2sin(s)) ,

$r'(s)=3s^2- {\color{DarkBlue} d(s^2sin(s))}$

$r'(s)=3s^2-{\color{DarkBlue} [s^2(cos(s))+2s(sin(s))]}$

Multiply the negative across:

r'(s)=3s^2-s^2cos(s)-2s(sin(s))

Factor out a common factor of s, and we reach the correct answer:

*****************************************************************

ANSWER

*****************************************************************

${\color{Red} r'(s)=s(3s-scos(s)-2sin(s)){\color{Red} }}$

Report an Error

Example Question #459 : Derivatives

Find the derivative of the function:

g(x)=3cos(x)-7sec(x)+5

Possible Answers:

$g'(x)=\frac{1}{3}[sin(x)+7sec(x)tan(x)]$

g'(x)=3sin(x)-7sec(x)tan(x)

g'(x)=3sec(x)+7sin(x)tan(x)

g'(x)=-3sin(x)-7sec(x)tan(x)

g'(x)=3sec(x)-7sin(x)tan(x)

Correct answer:

g'(x)=-3sin(x)-7sec(x)tan(x)

Explanation:

*****************************************************************

STEPS

*****************************************************************

Given:

g(x)=3cos(x)-7sec(x)+5

We can avoid the product rule by factoring the constants out before derivation:

g'(x)=3d(cos(x))-7(d(sec(x)))+d(5)

g'(x)=(3)(-sin(x))-7d(sec(x))+0

g'(x)=-3sin(x))-7(d(sec(x))

Taking the derivative of the secant function, we arrive at the answer:

*****************************************************************

ANSWER

*****************************************************************

${\color{Red} g'(x)=-3sin(x)-7sec(x)tan(x)}$

Report an Error

Example Question #81 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the function:

z(y)=y^4tan(y)

Possible Answers:

z'(y)=y^3[ysec^2(y)+4tan(y)]

$z'(y)=-\frac{1}{y^2}[ysec^2(y)+4tan(y)]$

$z'(y)=\frac{1}{y^2}[ysec^2(y)-4tan(y)]$

z'(y)=y^2[ysec^2(y)-4tan(y)]

$z'(y)=-\frac{1}{y^2}[ysec^2(y)-4tan(y)]$

Correct answer:

z'(y)=y^3[ysec^2(y)+4tan(y)]

Explanation:

*****************************************************************

STEPS

*****************************************************************

Given:

z(y)=y^4tan(y)

Establish the derivative using the product rule:

z'(u*v)=u(v')+v(u')

z'(y)=y^4(sec^2(y))+4y^3(tan(x))

Distribute y terms

z'(y)=y^4sec^2(y)+4y^3tan(x)

Note and factor out the greatest common factor, y^3 to get the answer:

*****************************************************************

ANSWER

*****************************************************************

${\color{Red} z'(y)=y^3[ysec^2(y)+4tan(y)]}$

Report an Error

Example Question #82 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the function:

s(r)=4sin(r)-3cos(r)

Possible Answers:

$s'(r)=\frac{3cos(r)+4sin(r)}{2}$

$s'(r)=-\frac{3cos(r)+4sin(r)}{2}$

$s'(r)=-\frac{4cos(r)+sin(r)}{2}$

s'(r)=3cos(r)+4sin(r)

s'(r)=4cos(r)+3sin(r)

Correct answer:

s'(r)=4cos(r)+3sin(r)

Explanation:

*****************************************************************

STEPS

*****************************************************************

Given:

s(r)=4sin(r)-3cos(r)

We can circumvent having to use the product rule by factoring out the constants before deriving the functions, like so:

$s'(r)=4{\color{Magenta} d(sin(r))}-3{\color{Teal} d(cos(r))}$

Now, we simply derive the trigonometric functions:

$s'(r)={\color{Magenta} 4cos(r)}-3{\color{Teal} (-sin(r))}$

Multiplying the negatives, we arrive at the correct answer:

*****************************************************************

ANSWER

*****************************************************************

${\color{Red} s'(r)=4cos(r)+3sin(r)}$

Report an Error

Example Question #83 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the function:

x=csc(y)cot(y)

Possible Answers:

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

x'=csc(y)(csc^2(y)+cot^2(y))

x'=-csc(y)(csc^2(y)+cot^2(y))

$x'=2csc(y)(csc^2(y)+\frac{1}{2}cot^2(y))$

x'=2csc(y)(csc^2(y)+cot^2(y))

Correct answer:

x'=-csc(y)(csc^2(y)+cot^2(y))

Explanation:

*****************************************************************

STEPS

*****************************************************************

Given:

x=csc(y)cot(y)

Set up the product rule:

${\color{Magenta} d(u*v)=u(v')+v(u')}$

x'=csc(y)(-csc^2(y))+cot(y)(-csc(y)cot(y))

x'=-csc^3(y)-csc(y)cot^2(y)

Factor out a greatest common factor -csc(y), and we reach the answer:

*****************************************************************

ANSWER

*****************************************************************

${\color{Red} x'=-csc(y)(csc^2(y)+cot^2(y))}$

Report an Error

Example Question #84 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the function:

$s(r)=\frac{sin^2(r)}{cos^2(r)}$

Hint: Try to re-arrange the function first, and you can reach the answer faster and easier!

Possible Answers:

s'(r)=sin(r)sin(r)

s'(r)=-2tan(r)tan(r)

$s'(r)=\frac{1}{sin(r)sin(r)}$

s'(r)=-sin(r)sin(r)

s'(r)=2tan(r)sec^2(r)

Correct answer:

s'(r)=2tan(r)sec^2(r)

Explanation:

*****************************************************************

STEPS

*****************************************************************

Given:

$s(r)=\frac{sin^2(r)}{cos^2(r)}$

With a little bit of re-arranging the problem, we can circumvent the quotient rule entirely

$s(r)=\frac{(sin(r))^2}{(cos(r))^2}$

$s(r)=(\frac{sin(r)}{cos(r)})^{2}$

Substituting in tangent, we get

s(r)=tan(r)^2

We can now derive

s(r)=tan(r)^2

If we apply the general product rule, and remember to use the chain rule, we get:

$s'(r)=2tan(r)*{\color{Blue} (chain)}$

We can find our chained term by deriving tangent:

${\color{Blue} d(tan(r))=sec^2(r)}$

Plug this result into our chain and we arrive at the correct answer:

*****************************************************************

ANSWER

*****************************************************************

${\color{Red} s'(r)=2tan(r)sec^2(r)}$

Report an Error

Example Question #85 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the function:

$h(g)=e^{cos(g)}$

Possible Answers:

$h'(g)=sin(g)e^{cos(g)}$

$h'(g)=2sin(g)e^{2sin(g)}$

$h'(g)=sin(g)e^{sin(g)}$

$h'(g)=-2sin(g)e^{2sin(g)}$

$h'(g)=-sin(g)e^{cos(g)}$

Correct answer:

$h'(g)=-sin(g)e^{cos(g)}$

Explanation:

*****************************************************************

STEPS

*****************************************************************

Given:

$h(g)=e^{cos(g)}$

Understand the derivative of the exponential:

$h'(g)=e^{cos(g)}{\color{Blue} *(chain)}$

To obtain the chained term, we must derive the compound function element, in this case, the cos function contained in the exponent

${\color{Blue} d(cos(g))=-sin(g)}$

Plug the -sin back in:

$h'(g)=e^{cos(g)}{\color{Blue} *-sin(g)}$

Pull the sin to the front, and we arrive at the correct answer:

*****************************************************************

ANSWER

*****************************************************************

${\color{Red} h'(g)=-sin(g)e^{cos(g)}}$

Report an Error

Example Question #86 : Derivative Rules For Sums, Products, And Quotients Of Functions

Calculate the derivative of $\frac{ln(x)}{x}$ .

Possible Answers:

$\frac{1}{x^2}$

$\frac{1-ln(x)}{x}$

$\frac{-ln(x)}{x^2}$

$\frac{1-ln(x)}{x^2}$

Correct answer:

$\frac{1-ln(x)}{x^2}$

Explanation:

We don't have a formula for taking the derivative of this expression, so we'll have to use the quotient rule, since we have a fraction of functions.

The quotient rule is $(\frac{f}{g})'=\frac{g'f-f'g}{g^2}$ .

Applying it to $\frac{ln(x)}{x}$ , we get

$\frac{x*\tfrac{1}{x}-ln(x)*1}{x^2}=\frac{1-ln(x)}{x^2}$

Our answer is $\frac{1-ln(x)}{x^2}$ .

Report an Error

Example Question #87 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of $x^2\ln x$ .

Possible Answers:

$x+\ln x$

$x+x\ln x$

$\ln x$

$x+2x\ln x$

$2x\ln x$

Correct answer:

$x+2x\ln x$

Explanation:

Using the product rule, (fg)'=f*g'+g*f' , the derivative of $x^2\ln x$ is

$x^2(\tfrac{1}{x})+2x\ln x=x+2x\ln x$

Final answer:

$x+2x\ln x$

Report an Error

View AP Calculus AB Tutors

Bhupinder
Certified Tutor

Panjab University, Bachelor of Engineering, Biotechnology. Hamburg University of Technology, Master of Science, Chemical Engi...

View AP Calculus AB Tutors

Sumit
Certified Tutor

Calcutta University, Bachelor of Science, Mathematics. Indian Statistical Institute, Master of Science, Mathematics.

View AP Calculus AB Tutors

Sharif
Certified Tutor

NCCU, Bachelors, Physics and Mathematics.

All AP Calculus AB Resources

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

Popular Subjects

Biology Tutors in Boston, French Tutors in Washington DC, English Tutors in Dallas Fort Worth, Computer Science Tutors in Atlanta, LSAT Tutors in Philadelphia, LSAT Tutors in Dallas Fort Worth, Chemistry Tutors in Philadelphia, Chemistry Tutors in Seattle, Spanish Tutors in New York City, Biology Tutors in Dallas Fort Worth

Popular Courses & Classes

SAT Courses & Classes in San Diego, ISEE Courses & Classes in Seattle, GRE Courses & Classes in Atlanta, ISEE Courses & Classes in Denver, GRE Courses & Classes in Denver, MCAT Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Philadelphia, LSAT Courses & Classes in Washington DC, SSAT Courses & Classes in Boston, SSAT Courses & Classes in Seattle

Popular Test Prep

SSAT Test Prep in Houston, MCAT Test Prep in Chicago, ACT Test Prep in Dallas Fort Worth, LSAT Test Prep in Atlanta, GMAT Test Prep in Miami, SSAT Test Prep in Phoenix, GMAT Test Prep in Seattle, ACT Test Prep in Phoenix, GMAT Test Prep in Washington DC, GRE Test Prep in Los Angeles

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 #457 : Derivatives

Example Question #458 : Derivatives

Example Question #459 : Derivatives

Example Question #81 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #82 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #83 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #84 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #85 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #86 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #87 : 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: