Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus AB : Derivatives of functions

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 : Derivatives Of Functions

The Second Fundamental Theorem of Calculus (FTOC)

Consider the function equation (1)

$F(x)=\int_a^xf(t)dt$ (1)

The Second FTOC states that if:

is continuous on an open interval .
is in .
and is the anti derivative of

then we must have,

F'(x)=f(x) (2)

Differentiate,

$f(x)=\sin(x^2)+\int_o^{x^2+1}\ln(t+1)dt$

Possible Answers:

$f'(x)=2x\left[\cos(x^2)+\frac{1}{x^2+1}\right]$

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

$f'(x)=2x\cos(x^2)+\ln(x^2+1)$

$f'(x)=2x\left[\cos(x^2)+\ln(x^2+2)\right]$

Correct answer:

$f'(x)=2x\left[\cos(x^2)+\ln(x^2+2)\right]$

Explanation:

Differentiate:

$f(x)=\sin(x^2)+\int_o^{x^2+1}\ln(t+1)dt$

$f'(x) = \frac{d}{dx}\sin(x^2)+\frac{d}{dx}\int_o^{x^2+1} \ln(t+1)dt$

Both terms must be differentiated using the chain rule. The second term will use a combination of the chain rule and the Second Fundamental Theorem of Calculus. To make the derivative of the second term easier to understand, define a new variable so that the limits of integration will have the form shown in Equation (1) in the pre-question text.

Let,

u= x^2+1

Therefore,

$\frac{du}{dx}=2x$

Now we can write the derivative using the chain rule as:

$f'(x) = 2x\cos(x^2)+\frac{du}{dx}\frac{d}{du}\left(\int_o^{u} \ln(t+1)dt\right)$

$f'(x) = 2x\cos(x^2)+2x\underset{Use_, FTOC}{ \underbrace{\frac{d}{du}\left(\int_o^{u} \ln(t+1)dt\right)}}$

Let's calculate the derivative with respect to in the second term using the Second FTOC and then convert back to .

$\frac{d}{du}\int_o^{u} \ln(t+1)dt = \ln (u+1) = \ln\left \{ (x^2+1)+1 \right \} = \ln(x^2+2)$

Therefore we have,

$f'(x)=2x\cos(x^2)+2x\ln(x^2+2)$

$f'(x)=2x\left[\cos(x^2)+\ln(x^2+2)\right]$

Report an Error

Example Question #1 : Derivatives Of Functions

Find the derivative.

12x^3

Possible Answers:

24x^2

36x^2

36x^3

Correct answer:

36x^2

Explanation:

Use the power rule to find the derivative.

$\frac{d}{dx}12x^3=36x^2$

Report an Error

Example Question #2 : Derivatives Of Functions

Find the derivative of the following:

f(x)=3x+2

Possible Answers:

None of the above

f'(x)=5

f'(x)=3

f'(x)=3x^2+2x

Correct answer:

f'(x)=3

Explanation:

To take the derivative you need to bring the power down to the front of the equation, multiplying it by the coefficient and then drop the power.

So:

f(x)=3x+2 becomes f'(x)=3

because the degree of "x" is just one, and once you multiply 3 by 1 you get 3 and drop the power of "x" to 0. The second term is just a constant and the derivative of any term is just 0.

Report an Error

Example Question #2 : Derivatives Of Functions

Find f'(x) :

$f(x)=\frac{2x}{x}$

Possible Answers:

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

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

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

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

f'(x)=0

Correct answer:

f'(x)=0

Explanation:

To do this problem you need to use the quotient rule. So you do

(low)(derivative of the high)-(high)(derivative of the low) all divided by the bottom term squared.

So:

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

Which, when simplified is:

$\frac{2x-2x}{x^2}=0$

Report an Error

Example Question #3 : Derivatives Of Functions

Find f'(x) :

(3x^3)(x)

Possible Answers:

f'(x)=9x^2

f'(x)=9x^3

f'(x)=12x^2

f'(x)=9x^3+3x^2

f'(x)=12x^3

Correct answer:

f'(x)=12x^3

Explanation:

This is the product rule, which is: (derivative of the first)(second)+(derivative of the second)(first)

So:

(9x^2)(x)+(1)(3x^3)=12x^3

Report an Error

Example Question #4 : Derivatives Of Functions

Find the derivative of the following:

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

Possible Answers:

f'(x)=1

f'(x)=x-1

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

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

f'(x)=0

Correct answer:

f'(x)=1

Explanation:

This is a combination of chain rule and quotient rule.

So:

$2x(\frac{1}{x})+\frac{0-1}{x^2}(x^2)=2-\frac{x^2}{x^2}$

Which when simplified you get:

f'(x)=2-1=1

Report an Error

Example Question #5 : Derivatives Of Functions

Find the derivative of the following:

f(x)=sin(x)+tan(x)

Possible Answers:

f'(x)=-cos(x)-sec^2(x)

f'(x)=cos(x)-sec^2(x)

f'(x)=cos(x)+sec^2(x)

f'(x)=-cos(x)+sec^2(x)

Correct answer:

f'(x)=cos(x)+sec^2(x)

Explanation:

This problem is just addition of derivatives using trigonometric functions.

So:

f'(x)=cos(x)+sec^2(x)

Report an Error

Example Question #24 : Ap Calculus Ab

Find the derivative:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The is a quotient rule using a trigonometric function.

So:

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

You can pull out an "x" and cancel it to get:

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

Report an Error

Example Question #6 : Derivatives Of Functions

Find the derivative:

$f(x)=x^{-10}$

Possible Answers:

$f'(x)=\frac{-10}{x^{11}}$

$f'(x)=\frac{10}{x^{11}}$

$f'(x)=\frac{-10}{x^9}$

$f'(x)=\frac{10}{x^9}$

Correct answer:

$f'(x)=\frac{-10}{x^9}$

Explanation:

This is the same concept as a normal derivative just with a negative in the exponent.

$f'(x)=-10(x^{-9})$

which becomes:

$f'(x)=\frac{-10}{x^9}$

Report an Error

Example Question #26 : Ap Calculus Ab

Calculate f'(x) :

$\small f'(x)={(x^2)}^2$

Possible Answers:

$\small f'(x)=4x^2$

$\small f'(x)=2(x^2)(2x)$

$\small f'(x)=(x^2)(2x)$

$\small f'(x)=3x^2$

Correct answer:

$\small f'(x)=2(x^2)(2x)$

Explanation:

This is a power rule that can utilize u-substitution.

So $\small f'(x)=(u)^2*u'$

where $\small u=x^2$

So you get:

$\small f'(x)=2u*u'$

Plug "u" back in and you get:

$\small f'(x)=2(x^2)(2x)$

Report an Error

View AP Calculus AB Tutors

Mohammed
Certified Tutor

University of Maryland-College Park, Bachelor of Science, Electrical Engineering. University of Maryland-College Park, Master...

View AP Calculus AB Tutors

Pankaj
Certified Tutor

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

View AP Calculus AB Tutors

Abhisek
Certified Tutor

University of Florida, Bachelor of Engineering, Computer Science.

All AP Calculus AB Resources

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

Popular Subjects

French Tutors in Phoenix, English Tutors in New York City, ISEE Tutors in Denver, SSAT Tutors in Seattle, Math Tutors in Chicago, LSAT Tutors in Los Angeles, ISEE Tutors in San Francisco-Bay Area, Computer Science Tutors in Denver, Biology Tutors in Atlanta, GMAT Tutors in Washington DC

Popular Courses & Classes

LSAT Courses & Classes in Los Angeles, MCAT Courses & Classes in New York City, GRE Courses & Classes in Dallas Fort Worth, ACT Courses & Classes in Seattle, SSAT Courses & Classes in Washington DC, Spanish Courses & Classes in Houston, Spanish Courses & Classes in Philadelphia, ACT Courses & Classes in Denver, LSAT Courses & Classes in Washington DC, GRE Courses & Classes in Chicago

Popular Test Prep

GMAT Test Prep in Seattle, ISEE Test Prep in Philadelphia, SSAT Test Prep in Washington DC, GMAT Test Prep in Dallas Fort Worth, MCAT Test Prep in Chicago, SAT Test Prep in New York City, SAT Test Prep in San Diego, ACT Test Prep in Denver, SSAT Test Prep in Dallas Fort Worth, MCAT Test Prep in Phoenix

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 : Derivatives of functions

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #1 : Derivatives Of Functions

Example Question #1 : Derivatives Of Functions

Example Question #2 : Derivatives Of Functions

Example Question #2 : Derivatives Of Functions

Example Question #3 : Derivatives Of Functions

Example Question #4 : Derivatives Of Functions

Example Question #5 : Derivatives Of Functions

Example Question #24 : Ap Calculus Ab

Example Question #6 : Derivatives Of Functions

Example Question #26 : Ap Calculus Ab

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: