We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : Differentiable Rate

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 #1 : How To Find Differentiable Of Rate

For the relation $\small x^2y+2x-y^2=x$ , compute $\small y'$ using implicit differentiation.

Possible Answers:

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

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

$\small \small \small \small y'=-\frac{2xy+1}{y(x^2-2y)}$

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

Correct answer:

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

Explanation:

Computing $\small y'$ of the relation $\small x^2y+2x-y^2=x$ can be done through implicit differentiation:

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

$\small \small \implies 2xy+x^2y'+2-2yy'=1$

Now we isolate the $\small y'$ :

$\small \small \small \iff x^2y'-2yy'=1-2-2xy=-2xy-1$

$\small \iff y'(x^2-2y)=-(2xy+1)$

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

Report an Error

Example Question #1 : How To Find Differentiable Of Rate

In chemistry, rate of reaction is related directly to rate constant .

$\frac{dC}{dt}=k$ , where is the initial concentration

Give the concentration of a mixture with rate constant $k=\frac{3}{sec}$ and initial concentration C(0)=40M , seconds after the reaction began.

Possible Answers:

$40e^{(-30)}$

30M

70M

$30e^{(-40)}$

Correct answer:

70M

Explanation:

This is a simple problem of integration. To find the formula for concentration from the formula of concentration rates, you simply take the integral of both sides of the rate equation with respect to time.

$\int \frac{dC}{dt} dt=\int (k)dt$

Therefore, the concentration function is given by

C=kt+C_0 , where C_0 is the initial concentration.

Plugging in our values,

C=3*10+40= 70M

Report an Error

Example Question #2901 : Calculus

and are related by the function $y=3x^{4}-x+5$ . Find $\frac{dy}{dt}$ at t=2 if x=1 and $\frac{dx}{dt}=6$ at t=2 .

Possible Answers:

$\frac{dy}{dt}=12$

$\frac{dy}{dt}=66$

$\frac{dy}{dt}=60$

$\frac{dy}{dt}=73$

$\frac{dy}{dt}=6$

Correct answer:

$\frac{dy}{dt}=66$

Explanation:

We will use the chain and power rules to differentiate both sides of this equation.

Power Rule:

$\frac{d}{dx}x^n=nx^{n-1}$

Chain Rule:

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

Applying the above rules to our function we find the following derivative.

$\frac{dy}{dt}=\frac{d}{dt}[3x^{4}-x+5]$

$\frac{dy}{dt}=12x^{3}\frac{dx}{dt}-1\frac{dx}{dt}$

at t=2 , $\frac{dx}{dt}=6$ and x=1 .

Therefore at t=2

$\frac{dy}{dt}=12(1)^{3}(6)-1(6)=72-6=66$

Report an Error

Example Question #4 : How To Find Differentiable Of Rate

Let $y = x^{xlnx-e}.$ Use logarithmic differentiation to find y'(e) .

Possible Answers:

Correct answer:

Explanation:

The form of log differentiation after first "logging" both sides, then taking the derivative is as follows:

$\frac{1}{y}y' = \frac{d}{dx}(lnf(x))$ which implies

$y' = f(x)\frac{d}{dx}(lnf(x))$

So:

$y'(e)=e^{elne-e}(ln^{2}e+2lne-\frac{e}{e})= 1(1+2-1)= 2$

Report an Error

Example Question #2 : How To Find Differentiable Of Rate

We can interperet a derrivative as $\frac{dy}{dx}= \lim_{\Delta \rightarrow 0} \frac{\Delta y}{\Delta x}$ (i.e. the slope of the secant line cutting the function as the change in x and y approaches zero) but these so-called "differentials" ( $\Delta x$ and $\Delta y$ ) can be a good tool to use for aproximations. If we suppose that $\frac{dy}{dx}=f'(x)$ , or equivalently dy=f'(x)dx . If we suppose a change in x (have a concrete value for $\Delta x$ ) we can find the change in with the afore mentioned relation.

Let f(x)= 3x^2 + x -1 . Find f'(3) and, given $\Delta x = 0.01,$ and find $\Delta y$ .

Possible Answers:

0.034

-0.19

0.1

0.19

1.9

Correct answer:

0.19

Explanation:

Taking the derivative of the function:

$f'(x)= \frac{d}{dx}[3x^2 + x -1]$

$\frac{dy}{dx}= 6x + 1$

Evaluating at x=3 :

$\frac{dy}{dx}|_{x=3} = 6(3) + 1$

$\frac{dy}{dx}|_{x=3} = 6(3) + 1=19$

Manipulating the equation:

dy =19dx

Allowing dx to be .01:

dy=19*0.01

dy=0.19

Which is our answer.

Report an Error

Example Question #5 : How To Find Differentiable Of Rate

Let $f(x)= \tan (\cos x)$ . Find $f'(\frac{\pi}{4})$ given $\Delta y = 0.3$ , find $\Delta x.$

Possible Answers:

$\approx 0.3422$

$\approx 0.2327$

$\approx -1.0736$

$\approx -0.3669$

$\approx 1.223$

Correct answer:

$\approx -0.3669$

Explanation:

First, we take the derivative of the function:

$f'(x)=\frac{d}{dx} [\tan (\cos x)]$

$=\sec^2(\cos x)*(-\sin x)$

$=-\sin x \sec^2(\cos x)$

evaluate the derivative at $x= \frac{\pi}{4}$

$=-\sin (\frac{\pi}{4}) \sec^2(\cos (\frac{\pi}{4}))$

$=-\frac{\sqrt{2}}{2} \sec^2(\frac{\sqrt{2}}{2})$

Manipulating the equation by solving for dy:

$dy=-\frac{\sqrt{2}}{2} \sec^2(\frac{\sqrt{2}}{2})dx$

Assuming dx = 0.3

$dy=-\frac{\sqrt{2}}{2} \sec^2(\frac{\sqrt{2}}{2})*(0.3)$

$\approx -0.3669$

Report an Error

Example Question #4 : Differentiable Rate

The find the change of volume of a spherical balloon that is growing from to 5.5in.

Possible Answers:

$50\pi\; cubic \;inches$

$5\pi\; cubic \;inches$

$25\pi\; cubic \;inches$

$10\pi\;cubic\;inches$

$100\pi\; cubic \;inches$

Correct answer:

$50\pi\; cubic \;inches$

Explanation:

This is a related rate problem. To find the rate of change of volume with respect to radius, we need to take the derivative of the volume of a sphere equation $(\frac{4}{3}\pi r^{3}).$

Then, we will plug in the relevant information. The initial radius will be substituted in for , and dr=0.5in , since that is the change from the initial to final radius of the balloon.

$dV=4\pi r^{2}dr.$

$dV=4\pi (5in)^{2}(0.5in)=50\pi in^{3}.$

Report an Error

Example Question #5 : How To Find Differentiable Of Rate

Find the rate of change of $y=4x^{3}-12x^{2}+6x+7$ at x=2 .

Possible Answers:

Correct answer:

Explanation:

To find the rate of change of a polynomial at a point, we must find the first derivative of the polynomial and evaluate the derivative at that point.

For this problem,

$y=4x^{3}-12x^{2}+6x+7$

the first derivative of this expression is

$\frac{dy}{dx}=12x^{2}-24x+6$

at x=2 the rate of change is

$\frac{dy}{dx}=12*(2)^{2}-24(2)+6=48-48+6=6$

Report an Error

View Calculus Tutors

Shakeel
Certified Tutor

View Calculus Tutors

Evgenii
Certified Tutor

View Calculus Tutors

Shubham
Certified Tutor

Jaypee University of Information and Technology, Bachelor of Technology, Computer Science.

All Calculus 1 Resources

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

Popular Subjects

GRE Tutors in Houston, Algebra Tutors in Phoenix, GMAT Tutors in Boston, LSAT Tutors in Philadelphia, GRE Tutors in Chicago, SAT Tutors in Seattle, GMAT Tutors in Dallas Fort Worth, SSAT Tutors in Denver, SAT Tutors in Boston, English Tutors in San Diego

Popular Courses & Classes

ISEE Courses & Classes in Washington DC, ACT Courses & Classes in San Diego, SSAT Courses & Classes in San Diego, SSAT Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Miami, SAT Courses & Classes in San Francisco-Bay Area, SAT Courses & Classes in Phoenix, GMAT Courses & Classes in Boston, MCAT Courses & Classes in Philadelphia, GMAT Courses & Classes in San Diego

Popular Test Prep

LSAT Test Prep in San Francisco-Bay Area, SSAT Test Prep in Atlanta, ACT Test Prep in Philadelphia, GMAT Test Prep in San Diego, LSAT Test Prep in Chicago, ACT Test Prep in Seattle, SAT Test Prep in New York City, ACT Test Prep in San Diego, GRE Test Prep in Los Angeles, GRE Test Prep in San Francisco-Bay Area

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 : Differentiable Rate

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #1 : How To Find Differentiable Of Rate

Example Question #1 : How To Find Differentiable Of Rate

Example Question #2901 : Calculus

Example Question #4 : How To Find Differentiable Of Rate

Example Question #2 : How To Find Differentiable Of Rate

Example Question #5 : How To Find Differentiable Of Rate

Example Question #4 : Differentiable Rate

Example Question #5 : How To Find Differentiable Of Rate

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: