find-the-derivative-of-the-function-using-the-definition-of-a-derivative-state-the-domain-of-the-f-2

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.
$f(t)=5 t-9 t^{2}$

Bryan Lynn · Accepted Answer

already. So again, we&#x27;re going to use the definition of the derivative, which is tthe e f Prime of X is equal to the limits as h approaches zero uh, the function X plus H minus f of X over h and in this instance, our function R f and they gave it in terms of tea is gonna be five t minus T squid. How does this going? Minus 90 square 90 square. Okay, so that means that f of tea plus age will be equal to five times tea plus H minus nine T plus age square. Okay. And that&#x27;s actually gonna be equivalent to, So we&#x27;ll distribute the 55 t plus five h have minus nine t squared, minus 18 th minus nine h squared. So we can plug both of these into our definition. And that would be the limit as h purchase zero five T plus five age minus nine. T squared minus 18. Th minus nine h squared all over. H, okay. And so this is gonna be equal to the limit as age approaches. Zero. Ah, negative. Nine h squared. Plus five minus 18. He times age all over h. And what I forgot to do the top. Of course. Subtract my f of X rights. I wrote my f of X plus h but we should also have a minus five T and an A plus mind t squared. And so that would have cancelled out five teas and the 90 squares, OK? And so we can&#x27;t see it take in a child. Right? So this age can come out and that one and so this is gonna be than equal to the limit as H approaches zero of negative nine H plus five minus 18 t. Okay, well, h goes to zero. This is simply going to be equal to your neighbor. 19 become zero, so it&#x27;s gonna be able to five minus 18 t. Okay, so this is our derivative f prime. And now we want to know what the domain is of each of these. So our function original function five T minus t squared. This is defined everywhere. So the domain is negative. Fate to infinity and as well as our derivatives. There&#x27;s nothing you can put in here that would make undefined. So again, the domain is negative. Infinity to infinity. All real numbers

Leon Druch · Answer

Okay here we have a function. F F T. I&#x27;m going to use capital T. So as I&#x27;m doing all this work, my lower case T. S don&#x27;t confuse you and look like plus times. I&#x27;m going to use capital T. So F F T. Is defined to be 2.5 times T squared plus six times T. The domain of this function. The domain of a function ah is in this case is the variables T. The domain would be all the values of T where this function is defined. Well, this function is defined for any number that you plug into T. So the domain is, you know, all the real numbers, the entire X axis. The next thing we want to do is we want to calculate uh F prime of T. The derivative of F with respect to T. Using uh the definition of derivative. So the definition of the derivative for F prime of T is going to be to limit of F of T plus delta T minus F of T. All divided by delta T. And we take the limit of this expression as delta T approaches zero. So F of T plus delta T. Wherever you see T. In the original function, we&#x27;re gonna uh plug in T plus delta T. So F F T plus Delta T. is going to be 2.5. And instead of times T squared OK, we are doing T plus delta T to the second Plus six times T. But remember right now we&#x27;re writing the expression for F of T plus delta T. Uh So instead of tea, we&#x27;re going to be writing in T. Plus delta T. So plus six times this variable is really plus six times to T. Plus delta T. So that takes care of the F. Of T. Plus delta T. Now we have to subtract F. Of T. Which is the 2.5 T square plus 60. So we&#x27;re going to subtract 2.5 T sq. And since we&#x27;re subtracting FFT and this entire expressionist FFT, we&#x27;re subtracting each of these terms. So I already wrote minus 2.5 times T squared. Now, since once again, since I&#x27;m subtracting FFT, I also had to subtract 2 60. So that&#x27;s why we write -16. Now all of this gets divided by delta T. And I forgot to write the little limit. Um but so we&#x27;ll just squeeze it in here. We want to take the limit of this whole expression as delta T approaches zero. So what I&#x27;m going to do next is I&#x27;m going to do T plus delta T to the second. Using algebra. If you think of foil basically this is a binomial. Um multiplying a binomial, you would really have T plus delta T. To the second is T. Plus delta T. Times itself. That&#x27;s what it means to raise it to the second. So T plus delta T. Times T. Plus delta T. You can use foil to multiply two by no mules. And I&#x27;m gonna start changing the color just to help make it look a little less cluttered here. So we have 2.5 and that has two times two T. Plus delta T squared. Well, T plus delta T square T. Plus delta T. Times itself is going to be T squared plus two times T times delta T. Plus delta T squared. So all of that in princes has to be multiplied by 2.5. Uh And then we have to do, I&#x27;m just going to bring this down for now and then I&#x27;m going to scroll down on the screen and will continue simplifying this expression. At this point. It looks really confusing, but a lot of things will drop out nicely. So plus six times T plus delta T minus 2.5 T. Times T squared minus 16. And all of that has to be divided by delta T. So let&#x27;s scroll down so that what we have in blue is near the top of this page so that we have some room to continue simplifying. And let&#x27;s change the green for now. So we want to take the limit of this entire expression. This delta T approaches zero. Remember what we&#x27;re trying to do? We&#x27;re calculating the derivative of the given function. Using the definition of the rib there. It&#x27;s a 2.5 has to be distributed as it multiplies each to three terms and Princes. So we&#x27;re going to have 2.5 T squared plus 2.5 times to T delta T. 2.5 times two is five. So plus five times T delta take Then the 2.5 is going to talk to delta T squared plus 2.5 delta T squared. And then plus Now six has to multiply these two terms. So plus 60 plus six times delta T. Ah And then we&#x27;re minus seeing 22.5 T squared. And we&#x27;re minus ng or subtracting 2 16. And all of this once again is over the delta take. All right, let&#x27;s move down a little bit more and bring what we have in green near the top. All right. This is where things will start simplifying a little bit because we&#x27;ll be able to combine some terms. So, the limit of this entire expression is delta T approaches zero. Well, let&#x27;s combine like terms. First of all. 2.5 T squared minus 2.5 T squared cancels. So that works out nice. Um we have plus 60. subtract 60 that cancels out lace. We have a five T Delta T. We have 2.5 delta T squared. And we have a six delta T. So, these are all unlike terms. So we&#x27;re gonna write all three of these five, she delta T Plus 2.5 delta T square. Uh and then plus six Delta T. And all of that needs to be divided by delta. Take Now each of these three terms in the numerator are being divided by delta T. So we can separate ah these additions in the numerator and take each one of these three different terms and individually divided by delta T. So five tee times delta T. Over delta T. The delta teeth would cancel out. You would have five T. Here 2.5. Uh let&#x27;s write down the plus time here. The 2.5 times delta T squared divided by delta T. Or delta T squared divided by delta T. Leads you just a delta T. Up here&#x27;s a 2.5 delta T. And then here we have a plus and then six delta T divided by delta T. Two. Delta teas will cancel and you&#x27;ll have just six. So now as we take the limit of this expression five T plus 2.5 times delta T. Plus six as delta T. Goes to zero. The delta T. Here is going towards zero out of these three terms. Uh This middle term to 2.5 delta T. Is the only term that has adult it T. So is the delta T approaches zero. Here&#x27;s a delta T. It&#x27;s approaching zero. So 2.5 uh times delta T. Is going to approach zero because if delta T. Is approaching 2.5 times delta T approaches zero. This five T. Is going to stay five T. Okay because it&#x27;s delta T. That approaches zero, not the T. So five T stays five T. Likewise two plus six days plus six. So when we take this limit, when we take the limit of this expression, five T plus 2.5 times delta T plus six, as delta T approaches zero, we end up with five T Plus six. So that is the derivative of the function FF. T. Okay, so this just moved down a little bit. Yeah, this is our answer for F prime of T. And we evaluated it or we found it by using the definition, the limit definition of the derivative. So at prime of T equals five T plus six. Uh If we went to the top of the problem and we found a derivative, the old fashioned way, it should be five T plus six. All right. Uh If we found a derivative of this 2.5 times T squared would be two times 2.5 uh times T. Which would be five T plus derivative 60 is six. So derivative of this is five T plus six. And that&#x27;s exactly what we found basically doing it. More complicated way using the definition of the derivative, the limit definition of the derivative last but not least. What is the domain of the F prime of T. The derivative function. Well, the derivative is F I&#x27;m sorry, the derivative is five times T plus six. This is defined for any real value of T. So basically, T could be any number. So the domain of F prime A. T once again is the entire X axis, any number on the X axis or basically any real number.

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.

Bryan Lynn · Answer

Okay, so we want to use the definition of the derivative to find it, and that definition is just a reminder. The limit as H approaches zero of the function X plus H minus the function at X over H and the function we will be working with is FX is equal to X squared minus two x cube. So that means that the function at expose each will be X plus H squared minus two times x plus h cubed is so if we distribute everything, this is going to be equal to X squared plus two x h plus h squared than now when this starts attracting, right, So it&#x27;s going to be minus two X cubed minus six x squared H minus six X h squared minus to age cube. I guess we can put both of these into our definition. Us. That means that this is gonna be equal to the limit as H approaches zero uh, X squared plus two X h plus H squared minus two x cubed minus six x squared. H minus six ex Each square minus to h Q. Do that. We need to subtract the f of X. Okay, so then that&#x27;s a B minus. X squared, a minus two ex cubes drawing a little line here to kind of separate that and this will be all over. H will we get some cancellations when we do this, right? So actually, that part is this last one. This minus two Excuse being subtracted to be added together. So then we get X squared minus X squared cancels out. There&#x27;s two x cubed Will counts a lot with this other to execute. I guess that simplifies things a little bit. And that will bring us to being equal to the limit. As H approaches zero of two x h plus h squared minus six x squared H minus six X h squared minus to age. Cute of her age. Well, so now we can take an H out of everything, right? So you can get rid of one here, one here, one here, here and here. And so we do that. This isn&#x27;t being equal to the limit. Has H approaches zero of two eggs plus H minus six X square minus six X age minus to age squared. OK, so if you put a jiggle zero into that, you end up with this being equal to two X minus six X squared. So this is our derivative. You know, we want to take a look at the domains. Well, the domain of that is defined everywhere, and so is the derivative. So for both of them, the domain will be all real numbers or negative Infinity to infinity.

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.

Problem

Find the derivative of the function using the def…

Problem 23 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.
$f(t)=5 t-9 t^{2}$

Answer

domain of the function $(-\infty, \infty)$
domain of the derivative $(-\infty, \infty)$

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James Stewart

Biocalculus Calculus for the Life Sciences

Anna Marie V.

Kayleah T.

Caleb E.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

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domain of the function $(-\infty, \infty)$domain of the derivative $(-\infty, \infty)$

Biocalculus Calculus for the Life Sciences

Anna Marie V.

Kayleah T.

Caleb E.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartBiocalculus Calculus for the Life Sciences

Anna Marie V.

Kayleah T.

Caleb E.

Michael J.

domain of the function $(-\infty, \infty)$
domain of the derivative $(-\infty, \infty)$

James Stewart

Biocalculus Calculus for the Life Sciences