Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

Create An Account

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1002 : Differential Functions

Find the first derivative of $y=3x^2\cdot sin(x^2+1)$

Possible Answers:

y'=6x(x^2cos(x^2+1)+sin(x^2+1))

y'=-6xcos(x^2+1)

y'=6xcos(x^2+1)

y'=12x^2cos(x^2+1)

y'=3x(xcos(x^2+1)+2sin(x^2+1))

Correct answer:

y'=6x(x^2cos(x^2+1)+sin(x^2+1))

Explanation:

We must use the product rule here, which says

$\frac{d}{dx}\left [f(x)\cdot g(x) \right ]=f(x)g'(x)+f'(x)g(x)$

Here, f(x)=3x^2, g(x)=sin(x^2+1)

So, $y'=3x^2\cdot \left [ sin(x^2+1)\right ]'+(3x^2)'\cdotsin(x^2+1)$

Now, as we differentiate each term, we see that we will need the chain rule for the derivative in the first term, which says

$\frac{d}{dx}\left [f(g(x)) \right ]=f'(g(x))\cdot g'(x)$

Applying the rule, as we continue to differentiate

$y'=3x^2\cdot \left [ cos(x^2+1)\cdot (x^2+1)'\right ]+6x\cdot sin(x^2+1)$

$y'=3x^2\cdot \left [ cos(x^2+1)\cdot 2x\right ]+6x\cdot sin(x^2+1)$

Simplifying, we get

$y'=6x^3\cdot cos(x^2+1)+6x\cdot sin(x^2+1)$

y'=6x(x^2cos(x^2+1)+sin(x^2+1))

Report an Error

Example Question #1003 : Differential Functions

Find the first derivative of y=(3x+1)^3

Possible Answers:

y'=9(3x+1)^2

y'=3(3x+1)^2

y'=3(3x+1)^4

y'=(3x+1)^2

y'=9(3x+1)^3

Correct answer:

y'=9(3x+1)^2

Explanation:

Here, we will need the chain rule, which says

$\frac{d}{dx}\left [f(g(x)) \right ]=f'(g(x))\cdot g'(x)$

Here, f(x)=x^3, g(x)=3x+1

Differentiating, we get

$y'=3(3x+1)^2\cdot 3$

y'=9(3x+1)^2

Report an Error

Example Question #1001 : Differential Functions

Find the first derivative of y=sin^2(x^2)

Possible Answers:

y'=2sin^2(x^2)

y'=2cos^2(x^2)sin(x^2)

y'=4xsin^2(x^2)

y'=4xsin(x^2)cos(x^2)

y'=4xcos^2(x^2)

Correct answer:

y'=4xsin(x^2)cos(x^2)

Explanation:

We need to use the chain rule TWICE, which says

$\frac{d}{dx}(f(g(h(x))))=f'(g(h(x)))\cdot g'(h(x))\cdot h'(x)$

Here,

f(x)=x^2, g(x)=sin(x), h(x)=x^2

Differentiating, gives us

$y'=2\cdot sin(x^2)\cdot cos(x^2)\cdot 2x$

Simplifying,

y'=4xsin(x^2)cos(x^2)

Report an Error

Example Question #1004 : Differential Functions

Find the first derivative of y=x^2e^x

Possible Answers:

y'=2e^x

y'=2x+e^x

y'=xe^x(x+2)

y'=2xe^x+e^x

y'=2xe^x

Correct answer:

y'=xe^x(x+2)

Explanation:

We need to use the product rule, which says

$\frac{d}{dx}\left [f(x)\cdot g(x) \right ]=f(x)g'(x)+f'(x)g(x)$

Differentiating gives us

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

y'=x^2e^x+2xe^x

Factoring, gives us

y'=xe^x(x+2)

Report an Error

Example Question #1005 : Differential Functions

Find the first derivative of $y=3x\cdot tan(3x)$

Possible Answers:

y'=27xsec^2(3x)tan(3x)

y'=3sec^2(3x)

y'=9xsec^2(3x)tan(3x)

y'=9xsec^2(3x)+3tan(3x)

y'=3xsec^2(3x)

Correct answer:

y'=9xsec^2(3x)+3tan(3x)

Explanation:

We need the product rule, which says

$\frac{d}{dx}\left [f(x)\cdot g(x) \right ]=f(x)g'(x)+f'(x)g(x)$

Differentiating gives

$y'=3(x\cdot (tan(3x))'+(x)'\cdot tan(3x))$

To differentiate the second term, we need the chain rule, which says

$\frac{d}{dx}(f(g(x)))=f'(g(x))\cdot g'(x)$

Continuing to differentiate,

$y'=3(x\cdot sec^2(3x)\cdot 3+1\cdot tan(3x))$

Simplifying,

$y'=3(3x\cdot sec^2(3x)+tan(3x))$

y'=9xsec^2(3x)+3tan(3x)

Report an Error

Example Question #1002 : Differential Functions

Find the first derivative of $y=\frac{x^2-3}{x+4}$

Possible Answers:

y'=2x

y'=x^2+8x+3

$y'=\frac{x^2+8x+3}{x+4}$

$y'=\frac{x^2+8x+3}{(x+4)^2}$

$y'=\frac{2x}{x+4}$

Correct answer:

$y'=\frac{x^2+8x+3}{(x+4)^2}$

Explanation:

We need to use the quotient rule here to differentiate, which says

$\frac{d}{dx}\left [ \frac{f(x)}{g(x)} \right ]=\frac{g(x)f'(x)-f(x)g'(x)}{(g(x))^2}$

Applying the quotient rule to differentiate,

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

$y'=\frac{(x+4)\cdot 2x-(x^2-3)\cdot 1}{(x+4)^2}$

Simplifying,

$y'=\frac{2x^2+8x-x^2+3}{(x+4)^2}$

$y'=\frac{x^2+8x+3}{(x+4)^2}$

Report an Error

Example Question #821 : How To Find Differential Functions

Find the first derivative of y=sin^3(5x+4)

Possible Answers:

y'=3sin^2(5x+4)

y'=15 sin^2(5x+4) cos(5x+4)

y'=15cos^2(5x+4)

y'=5cos^3(5x+4)

y'=15sin^2(5x+4)

Correct answer:

y'=15 sin^2(5x+4) cos(5x+4)

Explanation:

We need to use the chain rule TWICE, which says

$\frac{d}{dx}(f(g(h(x))))=f'(g(h(x)))\cdot g'(h(x))\cdot h'(x)$

Here,

f(x)=x^3, g(x)=sin(x), h(x)=5x+4

Applying the rule to differentiate,

$y'=3\cdot sin^2(5x+4) \cdot cos(5x+4) \cdot 5$

Simplifying,

y'=15 sin^2(5x+4) cos(5x+4)

Report an Error

Example Question #1002 : Differential Functions

Find the first derivative of y=x^3-x^2sin(x)

Possible Answers:

y'=3x^2-2xsin(x)cos(x)

y'=3x^2-2xcos(x)

y'=3x^2-2x-cos(x)

y'=x(3x+xcos(x)-2sin(x))

y'=3x^2-2xsin(x)

Correct answer:

y'=x(3x+xcos(x)-2sin(x))

Explanation:

We will need to use the product rule to differentiate the second term, which says

$\frac{d}{dx}\left [f(x)\cdot g(x) \right ]=f(x)g'(x)+f'(x)g(x)$

Applying to differentiate y,

$y'=3x^2-\left [ x^2(sin(x))'+(x^2)'sin(x)\right ]$

$y'=3x^2-\left [ x^2(-cos(x))+2xsin(x)\right ]$

y'=3x^2+x^2cos(x)-2xsin(x)

Factoring to simplify, gives

y'=x(3x+xcos(x)-2sin(x))

Report an Error

Example Question #2032 : Calculus

Find the derivative of y using implicit differentiation for the following function:

x=2xy+3y^2

Possible Answers:

$\frac{\mathrm{dy} }{\mathrm{d} x}=8y$

$\frac{\mathrm{dy} }{\mathrm{d} x}=\frac{1-2y}{2x+6y}$

$\frac{\mathrm{dy} }{\mathrm{d} x}=2x+6y$

$\frac{\mathrm{dy} }{\mathrm{d} x}=\frac{1+2y}{2x+6y}$

Correct answer:

$\frac{\mathrm{dy} }{\mathrm{d} x}=\frac{1-2y}{2x+6y}$

Explanation:

To solve for y' - the derivative of y - we must use implicit differentiation. All of the normal differentiation rules apply, but when we take the derivative of y with respect to x we must always include $\frac{\mathrm{dy} }{\mathrm{d} x}$ .

For the function given, when we take the derivative of both sides of the equation with respect to x, we get

$1=2y+2x\frac{\mathrm{dy} }{\mathrm{d} x}+6y\frac{\mathrm{dy} }{\mathrm{d} x}$

using the following rules:

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

Finally, solve for $\frac{\mathrm{dy} }{\mathrm{d} x}$ :

$1-2y=\frac{\mathrm{dy} }{\mathrm{d} x}(2x+6y)$

$\frac{\mathrm{dy} }{\mathrm{d} x}=\frac{1-2y}{2x+6y}$ .

Report an Error

Example Question #821 : How To Find Differential Functions

Determine if the piecewise function is differentiable:

$f(x)=\begin{Bmatrix} x^3-5x, x\leq 2\\ 7x-16, x>2 \end{Bmatrix}$

Possible Answers:

It is differentiable and continuous

It is continuous but not differentiable

It is neither continuous nor differentiable

It is differentiable but not continuous

Correct answer:

It is differentiable and continuous

Explanation:

Remember, for a function to be differentiable, it must be continuous and differentiable at all points.

Since both functions are smooth and continuous, we look at their behavior at their intersection at x=2

$f(x)=\begin{Bmatrix} x^3-5x, x\leq 2\\ 7x-16, x>2 \end{Bmatrix}$

For the first function,

f(2)=2^3-5*2= -2

For its derivative, we use the power rule:

f'(x)= 3x^2-5 , f'(2)=3*2^2-5=7

For the second function:

f(2)=7*2-16=-2

For the second function's derivative, we use the power rule:

f'(x)=7, f(2)=7

Since both the derivatives and the function values agree, this function is differentiable at all points.

Report an Error

View Calculus Tutors

Oluwawole
Certified Tutor

University of South Florida-St. Petersburg Campus, Bachelor of Engineering, Mechanical Engineering.

View Calculus Tutors

Joseph
Certified Tutor

Rutgers University-New Brunswick, Bachelor in Arts, Chemistry. Rutgers Graduate School of Biomedical Sciences, Current Grad S...

View Calculus Tutors

Mohammad
Certified Tutor

Rutgers University-New Brunswick, Bachelor of Science, Mathematics. Rutgers University-New Brunswick, Master of Science, Stat...

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

SSAT Tutors in Denver, French Tutors in Seattle, Computer Science Tutors in Phoenix, French Tutors in Los Angeles, ACT Tutors in Dallas Fort Worth, Spanish Tutors in Atlanta, French Tutors in Washington DC, ISEE Tutors in San Francisco-Bay Area, Biology Tutors in San Diego, GMAT Tutors in New York City

Popular Courses & Classes

ISEE Courses & Classes in Chicago, ACT Courses & Classes in San Diego, ACT Courses & Classes in Phoenix, GRE Courses & Classes in New York City, LSAT Courses & Classes in Seattle, ISEE Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in Denver, MCAT Courses & Classes in Boston, GRE Courses & Classes in Houston, SSAT Courses & Classes in Dallas Fort Worth

Popular Test Prep

SSAT Test Prep in Los Angeles, SAT Test Prep in Denver, SSAT Test Prep in Houston, GRE Test Prep in Phoenix, MCAT Test Prep in Miami, SAT Test Prep in Philadelphia, ACT Test Prep in San Diego, GMAT Test Prep in Denver, ACT Test Prep in Phoenix, SAT 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

Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #1002 : Differential Functions

Example Question #1003 : Differential Functions

Example Question #1001 : Differential Functions

Example Question #1004 : Differential Functions

Example Question #1005 : Differential Functions

Example Question #1002 : Differential Functions

Example Question #821 : How To Find Differential Functions

Example Question #1002 : Differential Functions

Example Question #2032 : Calculus

Example Question #821 : How To Find Differential Functions

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: