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 #1 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the following equation:

$h(x) = \sin(x)\cos(x)$

Possible Answers:

$\cos ^{2}(x)+\sin^{2}(x)$

$-\sin(x)\cos(x)$

$\cos ^{2}(x)$

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

None of the other answers

Correct answer:

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

Explanation:

Because this problem contains two functions multiplied together that can't be simplified any further, it calls for the product rule, which states that $\frac{d}{dx}(f(x)g(x)) = f{}'(x)g(x) + f(x)g{}'(x)$ .

By using this rule, we get the answer:

$\cos(x)\cos(x) + \sin(x)(-\sin (x))$

By simplifying, we conclude that the derivative is equal to

$\cos ^{2}(x)-\sin ^{2}(x)$ .

Report an Error

Example Question #8 : Derivative Rules For Sums, Products, And Quotients Of Functions

$\begin{align*}&\text{Calculate the derivative of the function}\\&-\frac{(114cos(19x))}{x^{20}}\end{align*}$

Possible Answers:

$-\frac{ (43320cos(19x))}{x^{21}}-\frac{ (43320sin(19x))}{x^{20}}$

$\frac{120}{x^{21}}- 361sin(19x)$

$\frac{(2280cos(19x))}{x^{21}}+\frac{ (2166sin(19x))}{x^{20}}$

$-\frac{(43320sin(19x))}{x^{21}}$

Correct answer:

$\frac{(2280cos(19x))}{x^{21}}+\frac{ (2166sin(19x))}{x^{20}}$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }cos(19x)\\&\text{and the second being: }\frac{1}{x^{20}}\\&\text{With the product scaled by }-114\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[cos(ax)]=-asin(ax)dx\\&d[x^a]=ax^{a-1}dx\\&\text{From the above we can continue:}\\&u=cos(19x);du=-19sin(19x)\\&v=\frac{1}{x^{20}};dv=-\frac{20}{x^{21}}\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&\frac{(2280cos(19x))}{x^{21}}+\frac{ (2166sin(19x))}{x^{20}}\end{align*}$

Report an Error

Example Question #9 : Derivative Rules For Sums, Products, And Quotients Of Functions

$\begin{align*}&\text{Take the derivative with respect to }x\text{ of the function}\\&y=-\frac{(380tan(17x))}{2^{(20x)}}\end{align*}$

Possible Answers:

$\frac{(380ln(2))}{2^{(20x)}}+ 340tan(17x)^{2} + 340$

$\frac{(7600\cdot (17tan(17x)^{2} + 17))}{2^{(20x)}}+\frac{ (129200tan(17x)ln(2))}{2^{(20x)}}$

$\frac{(7600tan(17x)ln(2))}{2^{(20x)}}-\frac{ (380\cdot (17tan(17x)^{2} + 17))}{2^{(20x)}}$

$\frac{(20ln(2)\cdot (6460tan(17x)^{2} + 6460))}{2^{(20x)}}$

Correct answer:

$\frac{(7600tan(17x)ln(2))}{2^{(20x)}}-\frac{ (380\cdot (17tan(17x)^{2} + 17))}{2^{(20x)}}$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }tan(17x)\\&\text{and the second being: }\frac{1}{2^{(20x)}}\\&\text{With the product scaled by }-380\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[tan(ax)]=\frac{adx}{cos^2(ax)}=(atan^2(ax)+a)dx\\&d[b^{ax}]=ab^{ax}ln(b)sx\\&\text{From the above we can continue:}\\&u=tan(17x);du=17tan(17x)^{2} + 17\\&v=\frac{1}{2^{(20x)}};dv=-\frac{(20ln(2))}{2^{(20x)}}\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&\frac{(7600tan(17x)ln(2))}{2^{(20x)}}-\frac{ (380\cdot (17tan(17x)^{2} + 17))}{2^{(20x)}}\end{align*}$

Report an Error

Example Question #381 : Derivatives

$\begin{align*}&\text{Calculate the derivative of the function}\\&56sin(2x)tan(16x)\end{align*}$

Possible Answers:

$2cos(2x)\cdot (896tan(16x)^{2} + 896)$

$112sin(2x)\cdot (16tan(16x)^{2} + 16) + 1792cos(2x)tan(16x)$

$56sin(2x)\cdot (16tan(16x)^{2} + 16) + 112cos(2x)tan(16x)$

$- 8cos(2x) - 224tan(16x)^{2} - 224$

Correct answer:

$56sin(2x)\cdot (16tan(16x)^{2} + 16) + 112cos(2x)tan(16x)$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }tan(16x)\\&\text{and the second being: }sin(2x)\\&\text{With the product scaled by }56\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[tan(ax)]=\frac{adx}{cos^2(ax)}=(atan^2(ax)+a)dx\\&d[sin(ax)]=acos(ax)dx\\&\text{From the above we can continue:}\\&u=tan(16x);du=16tan(16x)^{2} + 16\\&v=sin(2x);dv=2cos(2x)\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&56sin(2x)\cdot (16tan(16x)^{2} + 16) + 112cos(2x)tan(16x)\end{align*}$

Report an Error

Example Question #391 : Derivatives

$\begin{align*}&\text{Find the derivative of the function }\\&f(x)=\frac{(136ln(17x))}{x^{8}}\end{align*}$

Possible Answers:

$\frac{136}{x^{9}}-\frac{ 8}{x}$

$-\frac{1088}{x^{10}}$

$-\frac{ (18496ln(17x))}{x^{9}}-\frac{ 1088}{x^{9}}$

$\frac{136}{x^{9}}-\frac{ (1088ln(17x))}{x^{9}}$

Correct answer:

$\frac{136}{x^{9}}-\frac{ (1088ln(17x))}{x^{9}}$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }\frac{1}{x^{8}}\\&\text{and the second being: }ln(17x)\\&\text{With the product scaled by }136\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[x^a]=ax^{a-1}dx\\&d[ln(ax)]=\frac{dx}{x}\\&\text{From the above we can continue:}\\&u=\frac{1}{x^{8}};du=-\frac{8}{x^{9}}\\&v=ln(17x);dv=\frac{1}{x}\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&\frac{136}{x^{9}}-\frac{ (1088ln(17x))}{x^{9}}\end{align*}$

Report an Error

Example Question #212 : Computation Of The Derivative

$\begin{align*}&\text{Take the derivative with respect to }x\text{ of the function}\\&y=-182ln(16x)sin(4x)\end{align*}$

Possible Answers:

$-\frac{(728cos(4x))}{x}$

$56cos(4x) -\frac{ 13}{x}$

$-\frac{(728cos(4x))}{x}$ $-\frac{ (728sin(4x))}{x}- 11648cos(4x)ln(16x)$

$-\frac{ (182sin(4x))}{x}- 728cos(4x)ln(16x)$

Correct answer:

$-\frac{ (182sin(4x))}{x}- 728cos(4x)ln(16x)$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }ln(16x)\\&\text{and the second being: }sin(4x)\\&\text{With the product scaled by }-182\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[ln(ax)]=\frac{dx}{x}\\&d[sin(ax)]=acos(ax)dx\\&\text{From the above we can continue:}\\&u=ln(16x);du=\frac{1}{x}\\&v=sin(4x);dv=4cos(4x)\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&-\frac{ (182sin(4x))}{x}- 728cos(4x)ln(16x)\end{align*}$

Report an Error

Example Question #213 : Computation Of The Derivative

$\begin{align*}&\text{Calculate the derivative of the function}\\&-216\cdot 4^{(16x)}sin(12x)\end{align*}$

Possible Answers:

$-41472\cdot 4^{(16x)}cos(12x)ln(4)$

$- 2592\cdot 4^{(16x)}cos(12x) - 3456\cdot 4^{(16x)}sin(12x)ln(4)$

$- 41472\cdot 4^{(16x)}cos(12x) - 41472\cdot 4^{(16x)}sin(12x)ln(4)$

$192\cdot 4^{(16x)}ln(4) - 216cos(12x)$

Correct answer:

$- 2592\cdot 4^{(16x)}cos(12x) - 3456\cdot 4^{(16x)}sin(12x)ln(4)$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }4^{(16x)}\\&\text{and the second being: }sin(12x)\\&\text{With the product scaled by }-216\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[b^{ax}]=ab^{ax}ln(b)sx\\&d[sin(ax)]=acos(ax)dx\\&\text{From the above we can continue:}\\&u=4^{(16x)};du=16\cdot 4^{(16x)}ln(4)\\&v=sin(12x);dv=12cos(12x)\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&- 2592\cdot 4^{(16x)}cos(12x) - 3456\cdot 4^{(16x)}sin(12x)ln(4)\end{align*}$

Report an Error

Example Question #214 : Computation Of The Derivative

$\begin{align*}&\text{Find the derivative of the function }\\&f(x)=-119tan(6x)e^{(17x)}\end{align*}$

Possible Answers:

$12138tan(6x)e^{(17x)} - 2023e^{(17x)}\cdot (6tan(6x)^{2} + 6)$

$289e^{(17x)} - 42tan(6x)^{2} - 42$

$-2023e^{(17x)}\cdot (6tan(6x)^{2} + 6)$

$- 119e^{(17x)}\cdot (6tan(6x)^{2} + 6) - 2023tan(6x)e^{(17x)}$

Correct answer:

$- 119e^{(17x)}\cdot (6tan(6x)^{2} + 6) - 2023tan(6x)e^{(17x)}$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }e^{(17x)}\\&\text{and the second being: }-tan(6x)\\&\text{With the product scaled by }119\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[e^{ax}]=ae^{ax}dx\\&d[tan(ax)]=\frac{adx}{cos^2(ax)}=(atan^2(ax)+a)dx\\&\text{From the above we can continue:}\\&u=e^{(17x)};du=17e^{(17x)}\\&v=-tan(6x);dv=- 6tan(6x)^{2} - 6\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&- 119e^{(17x)}\cdot (6tan(6x)^{2} + 6) - 2023tan(6x)e^{(17x)}\end{align*}$

Report an Error

Example Question #215 : Computation Of The Derivative

$\begin{align*}&\text{Calculate the derivative of the function}\\&-\frac{(272cos(12x))}{3^{(12x)}}\end{align*}$

Possible Answers:

$\frac{(3264sin(12x))}{3^{(12x)}}+\frac{ (3264cos(12x)ln(3))}{3^{(12x)}}$

$\frac{(204ln(3))}{3^{(12x)}}- 192sin(12x)$

$\frac{(39168cos(12x)ln(3))}{3^{(12x)}}-\frac{ (39168sin(12x))}{3^{(12x)}}$

$-\frac{(39168sin(12x)ln(3))}{3^{(12x)}}$

Correct answer:

$\frac{(3264sin(12x))}{3^{(12x)}}+\frac{ (3264cos(12x)ln(3))}{3^{(12x)}}$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }\frac{1}{3^{(12x)}}\\&\text{and the second being: }cos(12x)\\&\text{With the product scaled by }-272\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[b^{ax}]=ab^{ax}ln(b)sx\\&d[cos(ax)]=-asin(ax)dx\\&\text{From the above we can continue:}\\&u=\frac{1}{3^{(12x)}};du=-\frac{(12ln(3))}{3^{(12x)}}\\&v=cos(12x);dv=-12sin(12x)\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&\frac{(3264sin(12x))}{3^{(12x)}}+\frac{ (3264cos(12x)ln(3))}{3^{(12x)}}\end{align*}$

Report an Error

Example Question #216 : Computation Of The Derivative

$\begin{align*}&\text{Find the derivative of the function }\\&f(x)=-112x^{17}ln(-4x)\end{align*}$

Possible Answers:

$119x^{16} -\frac{ 16}{x}$

$-1904x^{15}$

$- 1904x^{16}ln(-4x) - 112x^{16}$

$7616x^{16}ln(-4x) - 1904x^{16}$

Correct answer:

$- 1904x^{16}ln(-4x) - 112x^{16}$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }ln(-4x)\\&\text{and the second being: }x^{17}\\&\text{With the product scaled by }-112\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[ln(ax)]=\frac{dx}{x}\\&d[x^a]=ax^{a-1}dx\\&\text{From the above we can continue:}\\&u=ln(-4x);du=\frac{1}{x}\\&v=x^{17};dv=17x^{16}\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&- 1904x^{16}ln(-4x) - 112x^{16}\end{align*}$

Report an Error

View AP Calculus AB Tutors

Rachel
Certified Tutor

University of Arkansas at Little Rock, Bachelor in Arts, Mathematics.

View AP Calculus AB Tutors

Dossevi
Certified Tutor

cole Ste Genevive Versailles, Bachelor of Science, Mathematics. Institut National Polytechnique de Grenoble, Master of Scienc...

View AP Calculus AB Tutors

Michael
Certified Tutor

University of California-San Diego, Bachelors, Physics with specialization in Earth Science. University of California-San Die...

All AP Calculus AB Resources

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

Popular Subjects

GRE Tutors in New York City, Biology Tutors in Chicago, GRE Tutors in Chicago, SAT Tutors in Phoenix, LSAT Tutors in New York City, English Tutors in Los Angeles, French Tutors in Boston, Physics Tutors in Miami, ACT Tutors in New York City, Biology Tutors in Washington DC

Popular Courses & Classes

ISEE Courses & Classes in Denver, Spanish Courses & Classes in Seattle, Spanish Courses & Classes in Boston, SAT Courses & Classes in San Diego, SSAT Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Houston, MCAT Courses & Classes in Chicago, Spanish Courses & Classes in Los Angeles, GRE Courses & Classes in Boston, MCAT Courses & Classes in Phoenix

Popular Test Prep

SAT Test Prep in New York City, SAT Test Prep in Dallas Fort Worth, GRE Test Prep in Houston, ISEE Test Prep in San Diego, SAT Test Prep in San Diego, SAT Test Prep in Los Angeles, MCAT Test Prep in New York City, SSAT Test Prep in Washington DC, SAT Test Prep in Houston, MCAT Test Prep in Miami

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 #1 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #8 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #9 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #381 : Derivatives

Example Question #391 : Derivatives

Example Question #212 : Computation Of The Derivative

Example Question #213 : Computation Of The Derivative

Example Question #214 : Computation Of The Derivative

Example Question #215 : Computation Of The Derivative

Example Question #216 : Computation Of The Derivative

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: