find-the-derivative-of-the-function-using-the-definition-of-a-derivative-state-the-domain-of-the-f-8

Question

Find the derivative of the function using the definition of a derivative. State the domain of the function and the domain of its derivative.
$G(t)=\frac{4 t}{t+1}$

Bryan Lynn · Accepted Answer

Okay. So again, we want to use the function, the definition of the derivative of a function to solve the derivative and so that given a function, F uh, the derivative is defined as the limit as each approaches zero of the function evaluated at X plus H minus, the function evaluated at each over H I went ahead and made a typo says, minus the functioning violated at X. Okay. And so for this instance, we are given our function. F of tea is going to be equal to 40 over t plus one, and this means that are functioning. Evaluated at T plus H would be equal to 40 plus four age over t plus H plus one so we can plug these into our definition, which gives us the limit as H approaches zero of 40 plus four h over T plus age plus one minus four t over T plus one all that divided by H. Okay, so what we want to do is multiply the fracture on the left by t plus one over t plus one and the one on the right by Teeples H plus one over T plus age plus one. Okay, So when we do that, we will end up with the limit as each goes to zero of 40 square plus 40 plus for H T plus four H minus for a T squared minus for th minus 40 over t plus H plus one times t plus one times h. So now we&#x27;ll get some cancellations, right? So our four teas will cancel the four times T times h will cancel, and the four t squared will cancel. So all that&#x27;s left is on top is for H, and so this age will cancel with E H on the bottom. Okay, so it gives us the limit as age goes to zero of four over T plus age plus one times t plus one. So if we evaluate this at H equals zero, we get that this is equal to four over t plus one squared. Okay, so that is the solution for the derivative. And I want to take a look at the domain. So if we started the domain of the function here, the only problem is that we don&#x27;t want the denominator to be equal to zero. So in this case, our domain is going to be negative. Infinity to you. Negative one with negative one to infinity. Right? Or you could consider it all real numbers minus so all real numbers, but without negative one. And in fact, we get the exact same domain for the derivative, right? Anything except for negative one will evaluate here. So for both the function and the derivative, our domain is all real numbers, with the exception of the number negative one.

Mary Wakumoto · Answer

Alright, so here we have a function G F T which is one minus two T over three plus T. And we need to find its domain as well as di prima T and the domain of di prima T. And we&#x27;re going to use the formal definition of derivative domain by the way. Normally for any function is all real, unless you have some quote unquote bad excess, for example, we don&#x27;t want to divide by or in this case T. S we don&#x27;t want to divide by zero. So we&#x27;re gonna do is we&#x27;re gonna set three plus two equal to zero. Or we can say we cannot let it be zero. So therefore T cannot be -3. Therefore domain would be between minus infinity and -3 and -3 to infinity, basically all reels except keeping minus three. Alright, let&#x27;s go ahead now and solve for our derivative. Using definition of derivative. Okay, so G private E is equal to the limit As H goes to zero of G. Of T plus H minus G ft. All over age. Alright, so that&#x27;s our formal standard definition of derivative. Let&#x27;s go ahead and apply it then to our function. Okay, so that means for G every time I see a. T. It&#x27;s now T plus H for the first term. So I&#x27;ll get one minus twice T plus H over three plus T plus H and then minus GFT So minus um just GT as is. Okay, so that&#x27;s just all on top and on the bottom we still divide by eight. Okay, so this is super messy. one thing we can do is get a common denominator and try to clean it up that way. So let&#x27;s do that. So I&#x27;m going to basically um I need more room I think. So I&#x27;m going to kind of move over here so I have plenty of room to write this out. Okay so we are going to multiply the left fraction by three plus T. Over three plus T. This one here in the right fraction by this denominator which is really um three plus T. Plus H. So um okay so we&#x27;re gonna get a big mess but let&#x27;s do it. Okay so I&#x27;m going to get on the so I&#x27;m going to first look at this term and I&#x27;m going to multiply by three plus T over three plus T. That will give me one the same time. I&#x27;m going to distribute that too just to kind of clear that up. Okay so on the top we multiply by three plus T. Ah we&#x27;re going to have a common denominator so this whole big denominator will be three plus T plus H. Has three plus t. Okay but I need to know work with this one. I&#x27;m going to multiply top and bottom by three plus two plus eight so I will get minus 1 -2 T. Times three plus T plus H, wow. Right what a mess. So now I actually need to distribute unless I can recognize any terms that are going to cancel out. That will save me some work. Let&#x27;s see if there&#x27;s any obvious terms. It might be hard to do because I still need to kind of foil it all out. Um Let&#x27;s see. Uh Well I I think I&#x27;m just going to foil it out. There&#x27;s just so many terms. Okay so limit h well I&#x27;m gonna do it below because they&#x27;re really going to need a lot of room here and that was a close you&#x27;re going to see okay so here we go limit H. Goes to zero. We&#x27;re going to do a huge foil. So I&#x27;m going to take three times everything there and then I&#x27;m going to take so basically all the parts and then I&#x27;m going to do the same thing with T. So let&#x27;s keep track so I can have three minus 60 -6. H. That&#x27;s doing the in red now I&#x27;m going to do the same thing but T gets multiplied by all three parts. So I&#x27;ll get plus t -2 T. Squared -2 H. T. Alright so that took care of the left super foil. Uh This is crazy. Look how big this is. Okay now on the right we&#x27;re going to do the same trick. I&#x27;m gonna put this I&#x27;m gonna do one times each of the terms and then I will do the minus two T. Times E. To the terms we have a minus in front. So we have to distribute that as well. So that gives me -3 minus t minus H. Now do the minus two T. With double negative six plus two t. Times so plus 60 plus two t squared Plus two th that is amazing. Look at all those terms. Okay let&#x27;s look for cancelations. I get a plus two T squared and a minus two C squared. Goodbye. So we had a plus two th m minus two th another. Goodbye. Okay we have a plus 60 and a minus 60 and we have a plus 10 and minus T. This is exciting so much is going away and alright so then I think we&#x27;re left with they live it as H goes to zero. Let&#x27;s see. We have oh the three&#x27;s cancel ye so we&#x27;re down to um um minus seven H. So we have -7 h. All over. I forgot to divide but oh this is terrible I forgot to divide by age. Okay so I forgot that we&#x27;re still dividing by age here so I&#x27;m just going to if we divide a fraction by a single number that that never ends up on the bottom so I&#x27;m going to put the h down here we have an age down there so I will put it there and squish in the rest of the denominator three plus t. Alright now this is where it&#x27;s great because we need this final cancelation to have um hs canceled because we don&#x27;t want to be dividing by zero. Now I can finally plug in zero. That gives me -7/3. Three plus T. The H is zero and then times three plus T. So three plus t squared, wow. So that is G prime of T g prime of T. Okay, domain is the same as the original function. We just can&#x27;t have the denominator go to zero. So t cannot be minus three or you can think of it as minus infinity minus three and -3 to infinity as our domain. Okay. Woo lots of terms. All right. Have a great day. I hope that helped and see you soon.

Adnan Gill · Answer

so we have G of t is equal to one minus two t over three plus t. So first thing we need to do is finally domain. So the domain of dysfunction is t is not equal to negative three, so he can take all values except for negative three, cause that&#x27;s gonna make it nominate early voters you. Now, the next thing we want to do is find g prime of which fine definition is live. It each tends to zero g off T passage minus g of t over each. So you start solving for that so they have limited h tends to zero and gov t plus h means I&#x27;m gonna replace every TV fighting plus it. So why minus two times teeth plus age over three plus team plus H minus one minus two t over three plus t. So that is G off people s H minus GFT over each. So now, because thes are two fractions in the numerator. Therefore, before I can do anything else, I have to make sure that they both have the same denominator. So I&#x27;m gonna give post off them the same denominator, so I&#x27;m gonna have one minus two times T plus h over three plus team plus age. And what I&#x27;m going to do is I&#x27;m gonna multiply to the fraction in which the numerator and denominator are both going to be three dusty and I have minus one minus two tea or three first heat and I&#x27;m gonna multiply this with a fraction in which point, the numerator and denominator with Bt three plus 10 plus h over three plus t plus h. So the reason I&#x27;m doing that is to make sure that they both have the same denominators. Now that the denominator there same, I cannot combine those fractions and I can have only one denominator. So this would be limit. H tends to zero. Then I have each in the bottom. And now I will have one fraction in the numerator for which the common denominator would be three plus t plus h times three plus t. And now I&#x27;m gonna use foil to simplify this. So first is one times three outers is negative to t plus h times t. So that would be minus two t Sorry hunters. So outers would be, ah, one times t. So that would be a plus T and then innards will be minus two times T plus H times three. So that&#x27;s minus six times T plus H and then lasts would be minus two tee times T LeSage and then minus. I&#x27;m going to do the same thing here. Use foil. So forces One. Time Street, which is three outers, is one times t plus it, so that would be a plus team plus age and then dinners would be minus 60 and then lasts. Would be minus two tee times. T presage. So now I can actually simplify a few things here. So far thing. There&#x27;s a T and then minus t three and minus feet and there&#x27;s a T and there&#x27;s a minus t and then there&#x27;s a minus. Two tea time Steve Press agent minus tru TV Times tee presage so these can be eliminated. So now I have next line where I would say this is equal to live. It each tends to zero on. One more thing that we can do here is remember that age can be written as H over one, and since it&#x27;s in division, it can be changed into a multiplication with its reciprocal so Let&#x27;s force right what we have in the numerator. And then we must get back with one over each. So now we have three plus T plus h times three plus t And then we got times one over each. And now here we have minus 60 and then we have minus six h and then we have minus each and then we have plus 60. So this means we can eliminate the minus 60 with the plus 60. And then we can also combine the two terms attend agent them so this would be limit each tends to zero on this would be negative seven age over the plus T plus H times three plus t times one over h. And now the H in the new miniter can be cancelled with the age and the denominator. And at this point, we can now also substitute H is equal to zero. And when we do that, we get negative 7/3 plus t times the last team, or we can say that Gov t is equal to our Sergey. Prime of T is equal to negative 7/3 plus t squared And now for the domain of G prime of tea. Domain is also the same. T is not equal to negative. 30. So taken. Take all real values except for negativity.

Clarissa Noh · Answer

Hi, this is Claire. So when number 30 of Stuarts vile calculus books we&#x27;re gonna find the derivative isn&#x27;t the definition and state the domain of both the original and derivative wants you. It&#x27;s already wrote the definition of the derivative and the original function to make things faster. But instead of plugging explicit, each working on actually plucking tea, plus each were just substituting. Um, we&#x27;re substituting tea instead of Max. So it&#x27;s going to be G prime of tea calls women. That&#x27;s H approaches. So is going to be won over screw hump t plus beach subtracted by orginal questions. So one over a square root of team that&#x27;s all over each. When you simplify that, it becomes a woman, each ghosting squaring off T minus skirt of teeth claws each all over each time square root of T plus two each times squared off T. Then when you simplify that continuously, you&#x27;re going to get well in each approaches. Sarah and I got a beach over each times square root well, t times t plus slim this more not to play about one time more. One skirt of tea waas t plus beach. When you simplify that you get negative one over. It&#x27;s clear memory of tee times. T times were tough plus square root of tea. Then that&#x27;s flies into negative one team, too Negative. Three. House for our domain. For the original equation, it&#x27;s going to be, um, zero to infinity. We can&#x27;t we can&#x27;t use Brock. It&#x27;s because if we use zero on the bottom is gonna be undefined as well as infinity. Because one over in phonied infinity is zero. And that becomes undefined and the same for our f of prime well, G of crime. Miri. Instance. What? This is g of team and G prime tea and they both have the same dough means, yeah.

Ma. Theresa Alin · Answer

given GFT which is one over square root of tea. We need to find G prime of T. Using only derivative definition. Now by definition of derivatives we have G prime of T. This is equal to The limit as h approaches zero of G of T plus h minus guilty this all over each. And so from here we have limits As it approaches zero of 1 over the square root of T plus H -1 over the square of T. This times The reciprocal of a church is one over H. And then from here we want to combine the two fractions that&#x27;s going to be limit as h approaches zero of we have an LCD of square it of T plus each time squared of teeth. And the numerator should be squared of t minus squared of T passage this times one over each. And then from here we want to rationalize the numerator. We multiply the numerator and denominator by the conjugate that is squared of T plus squared of T plus H. This all over the same expression squared of T plus square it of T plus H. And so simplifying this. We have The limit as h approaches zero of we have squared of tea and then squared minus the square of the squared of sleepless age. This all over we have squares a sleepless age times a squared of tee, times H times the congregate squared of T plus squared of T plus H. I&#x27;m simplifying this. We have they met H approaches zero of the numerator becomes t minus T plus H this all over. Yes scared of T plus H. Time squared of T. Times H times squared of T plus squared of T passage. Simplifying further we have in here this becomes minus t minus H. And so this simplifies to negative H. Because the tea will cancel. And so we have limits S. H approaches zero of negative H. Over we have squares of C. Plus age times the square root of T. Times age times squared of T plus squared of T passage. And then from here we can cancel the H. Reduce it to one. And so we have limit as H approaches zero of negative of one over square it of T plus H. Times squared of T. Times the square of T plus the square of T plus H. and then evaluating at age we get negative one over Square of TP0 that&#x27;s squared of T. Times squared of T. Times we have squared of T. Plus scared of T. Plus zero. That&#x27;s just squared of T. And so we have negative one over scared of tea time scared of T. That&#x27;s just T. And then skirt of T. Plus scared of T. That&#x27;s two square it of T. And so we have negative one over to T squared of T. And so this is the derivative of the function for the domain of G. F T. And G. Prime of T. F G F T Is one over the square root of T. Then T. Is restricted. such that T must be greater than zero only. It cannot equal zero because T. Is or the squared of T is in the denominator where it cannot be zero there, so it&#x27;s only T greater than zero, or an interval notation. This is The interval from 0 to positive infinity, Where zero is not included. And if G prime of T is equal to negative one over to T squared of T, then it will have the same domain as the function because squared S. T. This is restricted. Such that team must be greater than zero because script of these in the denominator and the inside the squares must be positive. And so the domain must be The same 0 to positive infinity. zero is not included.

Adnan Gill · Answer

So we have a function. GFT is equal to one over the square of tea. So the domain off this function is oil numbers which are non zero and positive. So the domain of dysfunction would be from zero to infinity. Now we wanna find G prime of team, which would be limit. Each tends to zero G t plus H minus G off t divided by each. So that would be limit each 10 studio one, over square off T plus each minus one over square Gourcuff tea. And the whole thing is divided by each. So this is a subtraction of fractions. And before we can proceed, we want to make sure that they have the same denominator. So we&#x27;re gonna multiply the numerator and denominator off the second function. Second infraction by squared off T or plus each and this 1st 1 million a multi like the numerator and denominator by square Look of tea. So we get this will be limit each tense Does your squaring off t over square root of t times the square root of keep lesage minus the sweater. Tough team plus h over a squared off t times the square root of T plus age and the whole thing is divided by each. So now that the new monitor has the body fractions in the human aircraft, same denominator, we can just combine them. So this becomes limit. Each tends to zero squared off T minus squared off T plus h over squared off T times the square root of T plus each. And the whole thing is divided by each now each his actually fraction, which is a job for one. And this division will change into the multiplication by its reciprocal. So we can write that this would be called limit. Each tends to zero squared off T minus 28 off T plus H over swelling of tee times quicker 50 plus age times, one over each. So this age just gets part becomes part of the denominator. So we have limited stands to zero squared off t minus square of T plus H or H times, square of t times, the square root of tea passage. Now, you know what we need to do here is ah, a process called rationalization so hard this ass organization work. What we need to do is just remember that e minus B times a plus B is equal to on a squared minus B squared. So we say that is equal to the square root of tea and B is equal to the square root of tea. Presage. Then we can rationalize this by multiplying this whole fraction by a squared off T plus squared off key passage and also defined by school of T plus square of T plus each. So that&#x27;s what we&#x27;re gonna do next. So you know when we say it&#x27;s where minus piece where that&#x27;s gonna help us simplify this. So we have limit. Each tends to zero. So the human ITER not a squid of tea My nest sweated off key. Plus each times squared off t plus the square of passage divided by each times. Square it off tee times keep us age times the square root of T plus the square it off key passage. So this is becoming a little bit off a mess, but we clear up the mess in a second. So then this would be limited h turns to zero. So this is a squared minus be split and then over each times the square root of T times the square root of T plus H and then times quit of T plus the square root of T plus h. So now if we simplify the numerator if used distributed property, we get limit. Each tends to zero with a T minus T minus h over each times the square root of T times the square root of tea, passage times, sweating of team, plus the swearing off T plus H and now the T minus T. They cross out and and also now we have negative H over each, so that would also eliminate. So then we can say this is equal to live it each tends to zero, and now what we have in the A new manager is just a negative one. And in the denominator, we have splintered off the times work of T plus h times, square it off T plus squared off T plus teach. And now we can substitute H is equal to zero, so this would give us next one over square, tough tee times square of teeth and then times square of T plus square root of team. So this would people to negative one or a T times to square it off teeth. Or, in other words, we would say this is equal to negative one over to T squared off. So attack is the derivative off the function now for the domain. Now again, there&#x27;s a square root. So what that means is that the values must be greater than zero. You can&#x27;t have two equal to zero. So the domain off the derivative will also be from zero to infinity.

Problem

Find the derivative of the function using the def…

Problem 29 Medium Difficulty

Find the derivative of the function using the definition of a derivative. State the domain of the function and the domain of its derivative.
$G(t)=\frac{4 t}{t+1}$

Answer

domain of the function $\mathrm{R}-[1]$
domain of the derivative $\mathrm{R}-[1]$

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James Stewart

Biocalculus Calculus for the Life Sciences

Kayleah T.

Caleb E.

Samuel H.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

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domain of the function $\mathrm{R}-[1]$domain of the derivative $\mathrm{R}-[1]$

Biocalculus Calculus for the Life Sciences

Kayleah T.

Caleb E.

Samuel H.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartBiocalculus Calculus for the Life Sciences

Kayleah T.

Caleb E.

Samuel H.

Michael J.

domain of the function $\mathrm{R}-[1]$
domain of the derivative $\mathrm{R}-[1]$

James Stewart

Biocalculus Calculus for the Life Sciences