Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus BC : Computation of Derivatives

Study concepts, example questions & explanations for AP Calculus BC

Create An Account

All AP Calculus BC Resources

2 Diagnostic Tests 86 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

← Previous 1 2 3 4 5 6 7 8 Next →

Example Question #11 : Chain Rule And Implicit Differentiation

$\frac{d}{dx } \tan ^ 3 x$

Possible Answers:

$1 + \tan ^2 x$

$3 \tan ^2 x + 3 \tan ^4 x$

$2\tan^2 x$

$3 \tan ^2 x$

Correct answer:

$3 \tan ^2 x + 3 \tan ^4 x$

Explanation:

According to the chain rule, $f(g(x)) ' = f'(g(x))\cdot g'(x)$ . In this case, f(x) = x^3 and $g(x) = \tan x$ . The derivative is $3 (\tan x) ^2 \cdot( 1 + \tan ^2 x ) = 3 \tan ^2 x (1 + \tan ^2 x ) = 3 \tan ^2 x + 3 \tan ^4 x$

Report an Error

Example Question #11 : Chain Rule And Implicit Differentiation

$\frac{d}{dx } 4(x-2)^2$

Possible Answers:

8x - 16

-16(x-2)

Correct answer:

8x - 16

Explanation:

According to the chain rule, $f(g(x)) ' = f'(g(x)) \cdot g'(x)$ . In this case, f(x ) = 4x^2 and g(x) = x-2 . f'(x) = 8x and g'(x) = 1 .

The derivative is $8(x-2) \cdot 1 = 8x -16$

Report an Error

Example Question #11 : Chain Rule And Implicit Differentiation

$\frac{d}{dx} e^{\sin x}$

Possible Answers:

$- \cos x \cdot e ^ {\sin x }$

$\cos x \cdot e^ {\sin x}$

$e ^{ \cos x }$

$e ^ {\sin x }$

Correct answer:

$\cos x \cdot e^ {\sin x}$

Explanation:

According to the chain rule, $f(g(x)) ' = f'(g(x)) \cdot g'(x)$ . In this case, f(x) = e^x and $g(x) = \sin x$ . Since f'(x) = e^x and $g'(x) = \cos x$ , the derivative is $e ^ {\sin x } \cdot \cos x$

Report an Error

Example Question #11 : Chain Rule And Implicit Differentiation

$\frac{\mathrm{d} }{\mathrm{d} x} 3 ^ {x^2 }$

Possible Answers:

$2x \ln 3$

$3^ {2x} \cdot \ln(3)$

$3^{x^2} (\frac{x^2}{3} + 2x \ln 3)$

$3^{x^2} \cdot 2x \cdot \ln(3)$

Correct answer:

$3^{x^2} \cdot 2x \cdot \ln(3)$

Explanation:

According to the chain rule, $f(g(x)) ' = f'(g(x)) \cdot g'(x)$ . In this case, f(x) = 3^x and g(x) = x^2 . Since $f'(x) = 3^ {x} \cdot \ln(3)$ and g'(x) = 2x , the derivative is $3^{x^2 } \cdot \ln(3 ) \cdot 2x$

Report an Error

Example Question #11 : Chain Rule And Implicit Differentiation

$\frac{\mathrm{d} }{\mathrm{d} x} \enspace \frac{1}{4} \cdot 2^{4x-1}$

Possible Answers:

$\frac{\ln(2)}{4}$

$\frac{2^{4x-1}\cdot \ln (2)}{2}$

$\frac{2^{4x-1}}{4}$

$2^ {4x-1} \cdot \ln(2)$

Correct answer:

$2^ {4x-1} \cdot \ln(2)$

Explanation:

According to the chain rule, $f(g(x)) ' = f'(g(x)) \cdot g'(x)$ . In this case, $f(x) = \frac{1}{4} \cdot 2^x$ and g(x) = 4x-1 . Here $f'(x) = \frac{1}{4} \cdot 2^x \cdot \ln(2)$ and g'(x) = 4 . The derivative is:

$\frac{1}{4} \cdot 2 ^{4x-1} \cdot \ln(2) \cdot 4 = 2^{4x-1} \cdot \ln(2)$

Report an Error

Example Question #421 : Ap Calculus Bc

Given the relation y^2=x^2+y , find $\frac{dy}{dx}$ .

Possible Answers:

$\frac{2x}{y}$

$\frac{y}{x}$

$\frac{x}{y-1}$

$\frac{2x}{2y-1}$

Correct answer:

$\frac{2x}{2y-1}$

Explanation:

We begin by taking the derivative of both sides of the equation.

$\frac{d}{dx}(y^2)=\frac{d}{dx}(x^2+y)$ .

2yy' = 2x +y' . (The left hand side uses the Chain Rule.)

2yy' -y' = 2x .

(2y-1)y'=2x

$y' = \frac{2x}{2y-1}$

$\frac{dy}{dx} = \frac{2x}{2y-1}$ .

Report an Error

Example Question #422 : Ap Calculus Bc

Given the relation x^2+xy^2=y , find .

Possible Answers:

$\frac{2x}{2x+y}$

$\frac{x+y}{y^2}$

None of the other answers

$\frac{2x+y^2}{-2x+1}$

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

Correct answer:

$\frac{2x+y^2}{-2x+1}$

Explanation:

We can use implicit differentiation to find . We being by taking the derivative of both sides of the equation.

$\frac{d}{dx}(x^2+xy^2)=\frac{d}{dx}(y)$

$\frac{d}{dx}(x^2)+\frac{d}{dx}(xy^2)=y'$

$2x +[x \times 2y' + y^2 \times 1] = y'$ (This line uses the product rule for the derivative of xy^2 .)

2x +2xy' +y^2 = y'

2x+y^2 = -2xy' + y'

2x+y^2 = (-2x+1)y'

$\frac{2x+y^2}{-2x+1} = y'$

Report an Error

Example Question #11 : Chain Rule And Implicit Differentiation

If $f(x) = e^{e^x}$ , find f'(1) .

Possible Answers:

$e^{e+1}$

e^e

e^2

Correct answer:

$e^{e+1}$

Explanation:

Since we have a function inside of a another function, the chain rule is appropriate here.

The chain rule formula is

$(h(g(x)))' = h'(g(x))\times g'(x)$ .

In our function, both h(x), g(x) are e^x

So we have

$f'(x) = (h(g(x)))' = e^{e^x} \times e^x = e^{e^x+1}$

and

$f'(1)=e^{e+1}$ .

Report an Error

Example Question #11 : Chain Rule And Implicit Differentiation

Find $\frac{\mathrm{d} y}{\mathrm{d} x}$ of the following:

xe^y+y^2+x^2=y

Possible Answers:

e^y+xe^y+2y+2x

$\frac{-e^y-2x}{xe^y+2y}$

$\frac{-e^y-2x}{xe^y+2y-1}$

$\frac{-e^y-2x}{xe^y+2y+1}$

Correct answer:

$\frac{-e^y-2x}{xe^y+2y-1}$

Explanation:

To find $\frac{\mathrm{d} y}{\mathrm{d} x}$ we must use implicit differentiation, which is an application of the chain rule.
Taking $\frac{\mathrm{d} }{\mathrm{d} x}$ of both sides of the equation, we get

$\frac{\mathrm{d} y}{\mathrm{d} x}=e^y+xe^y\frac{\mathrm{d} y}{\mathrm{d} x}+2y\frac{\mathrm{d} y}{\mathrm{d} x}+2x$

using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} ax=a$

Note that for every derivative with respect to x of a function with y, the additional term dy/dx appears; this is because of the chain rule, where y=g(x), so to speak, for the function it appears in.

Using algebra, we get

$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{-e^y-2x}{xe^y+2y-1}$

Report an Error

Example Question #21 : Chain Rule And Implicit Differentiation

Find :

$f=\gamma \cos(x^2)$ , where $\gamma$ is a constant.

Possible Answers:

$2x\gamma \cos(x^2)$

$-2x\gamma \sin(x^2)$

$-\gamma \sin(x^2)$

$2x\gamma \sin(x^2)$

$-2x\sin(x^2)$

Correct answer:

$-2x\gamma \sin(x^2)$

Explanation:

The derivative of the function is equal to

$f'=-2x\gamma \sin(x^2)$

and was found using the following rules:

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

The constant may seem intimidating, but we treat it as another constant!

Report an Error

← Previous 1 2 3 4 5 6 7 8 Next →

View AP Calculus BC Tutors

Shakeel
Certified Tutor

View AP Calculus BC Tutors

Erik
Certified Tutor

University of Illinois at Urbana-Champaign, Bachelor of Science, Physics. University of California-San Diego, Master of Scien...

View AP Calculus BC Tutors

Pankaj
Certified Tutor

University of Delhi, Electrical Engineer, Electrical Engineering. Columbia University in the City of New York, Master of Arts...

All AP Calculus BC Resources

2 Diagnostic Tests 86 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Spanish Tutors in Houston, Algebra Tutors in Chicago, SAT Tutors in San Diego, SAT Tutors in Atlanta, English Tutors in Phoenix, Computer Science Tutors in Atlanta, Algebra Tutors in New York City, MCAT Tutors in Atlanta, Physics Tutors in Dallas Fort Worth, MCAT Tutors in Seattle

Popular Courses & Classes

MCAT Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Atlanta, GRE Courses & Classes in San Diego, SSAT Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in Atlanta, SSAT Courses & Classes in Seattle, ACT Courses & Classes in Miami, SSAT Courses & Classes in Denver, LSAT Courses & Classes in Washington DC, GRE Courses & Classes in Seattle

Popular Test Prep

LSAT Test Prep in New York City, ACT Test Prep in Miami, GRE Test Prep in San Diego, SSAT Test Prep in Denver, LSAT Test Prep in Miami, SSAT Test Prep in Atlanta, SAT Test Prep in New York City, SAT Test Prep in Phoenix, ACT Test Prep in San Francisco-Bay Area, ACT 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

AP Calculus BC : Computation of Derivatives

Study concepts, example questions & explanations for AP Calculus BC

All AP Calculus BC Resources

Example Questions

Example Question #11 : Chain Rule And Implicit Differentiation

Example Question #11 : Chain Rule And Implicit Differentiation

Example Question #11 : Chain Rule And Implicit Differentiation

Example Question #11 : Chain Rule And Implicit Differentiation

Example Question #11 : Chain Rule And Implicit Differentiation

Example Question #421 : Ap Calculus Bc

Example Question #422 : Ap Calculus Bc

Example Question #11 : Chain Rule And Implicit Differentiation

Example Question #11 : Chain Rule And Implicit Differentiation

Example Question #21 : Chain Rule And Implicit Differentiation

All AP Calculus BC Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: