find-the-derivative-of-the-function-using-the-definition-of-a-derivative-state-the-domain-of-the-f-6

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.
$g(x)=\sqrt{1+2 x}$

Bryan Lynn · Accepted Answer

Okay, so here we want to use the definition of the derivative which states that the definition of a derivative where the function is f of X, that&#x27;s gonna be equal to the limit as H approaches zero of the function evaluated at express age minus the function evaluated at X over H. And so for prom 27 um, defines our function is G of X vel McGee is sort of relevant being equal to the square root of one plus two x. And so this means that if we evaluate this function at X plus H, that&#x27;s going to be equal to the square root of one plus two times X plus h. We can plug both of these into our formula here, and we will get the limits as H approaches zero of the square root of one plus two x plus two age. So here just distributed that too minus the square root of one plus two x over h okay. And so that we can multiply both the top and the bottom. Bye. The square root of one plus two x plus two h plus the square root of one plus two X is gonna be to try get rid of some of these square roots. Can we have to do the same in the bottoms that we&#x27;re really just multiplying it by one. And once we do that, we end up with the limit as age of purchase zero of one plus two x plus two H minus one minus two X over h times the square root of one plus two x plus two h plus thes queer of one plus two acts. Okay. And then we get some cancellations. Right. So, uh, the one and the two X will both go away. Yes, we have just the two h on top. And so the H is going to cancel with the age below. And so this is equal to the limit. As H approaches zero of two over in course of H has had to zero this to H is also gonna go away. So we get to over two times the square root. So times the square one plus two x and I have skipped a little step here. Right? So we&#x27;re evaluating it at H equals zero house. We could remove this limit piece. Okay. And then the twos, they&#x27;re gonna cancel out, right? So we get that these twos cancel. And so the answer. Then it&#x27;s just gonna be one over the square root of one plus two x. Okay. And so now this one, right? We can&#x27;t divide by zero. So we do, in fact, have not a full domain. And in fact, we can&#x27;t. We can&#x27;t take, um, the square root of a negative number if we&#x27;re gonna work in reald numbers and so ex would need to be greater than negative 1/2 for the derivatives of the domain of the derivative is going to be negative 1/2 or larger, and then for the the function g of X. Well, this time we&#x27;re not worried about dividing by zero, but we still don&#x27;t want to be negative in the square root. So the domain here is gonna be negative 1/2 but including the negative 1/2 because it can equal zero ah to infinity.

Clarissa Noh · Answer

I think this is Clara. So when Number 26 of Stuart Spiral Calculus book. So we&#x27;re gonna find the dirt, but of using the definition state the domain of both the original Linda, you&#x27;re a bit of function. It&#x27;s already wrote the definition of the turbine, it and the origin equation above to make things faster. So for here we have, uh, we&#x27;re just gonna plug in X plus h into the origin equation. So that&#x27;s gonna give us limit. As each approaches, Ciro, exclude each plus, where were you? Thanks, Miss H minus X plus square E. Thanks all over each. And then if we simplify that further, we&#x27;re gonna get limit as future approaches. So each plus square right, uh, necks was beach minus square it, Lex, all over each. And that&#x27;s gonna give us limit. As each approaches zero, we&#x27;re gonna split this into two parts, so h over each gives us one, and then we&#x27;re gonna add so square. Marie, huh? Explicit age subtracted by square of X all over h. And we&#x27;re gonna simplify that further to get one plus limit a cz age approaches. Ciro. That&#x27;s plus H minus X over inch square. uh, each plus x plus square root of X, and when you simplify it, continue asleep. You got one minute. One plus limit of each approach is at zero one over square root square, X plus H plus square of axe. Then that will be cool. One plus one over two square root. Oh, that&#x27;s that&#x27;s your final answer for your domains. Where your original equation us with Lex, the domain of the function. It can&#x27;t be a negative, since it&#x27;s a square root. So zero to negative infinity. We could include Ciro. And for the domain of the derivative, the domain is going to be zero and negative Infinity. Unlike the original, we can&#x27;t put zero for the denominator or also be undefined. So we&#x27;re not. We&#x27;re not gonna amuse a bracket. We&#x27;re going to use that prevent disease

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.

Clarissa Noh · Answer

Hi. This is Claire. So in number 28 storage by calculus book. So we&#x27;re gonna find the derivative using the definition in state the domain of both the original and Derby Devotion. So here already wrote the orginal equation, And I also, um, both the definition of the derivative. So what, we&#x27;re gonna first do this plugging X plus eight, But the original equation. So it&#x27;s gonna be cold, woman, you know, seeing each approaches. Zero next. Once square minus one over to, uh, thanks. Once beach minus three more gonna subtract up by the original equations was next square minus one over two weeks, minus three all over each. We simplify this, it&#x27;s gonna become super long, so it&#x27;s gonna be We&#x27;ll start here when it does, each approaches. So we&#x27;re gonna get two ex cute waas for that square. Each plus two x each square minus two x minus three square minus six. Next, each minus three inch square plus three minus two X cube minus two that square kinds age waas three X square plus two minutes. Wants to teach minus three. Could the all over huge Klein&#x27;s q X waas to beach minus serene times two cups minus three I&#x27;ve only simplify this. Where can I get then? See Bones four score minus six bucks My nist to square waas too. Oh, movers two x plus three square. And then the final answer is to x square. My nose fits. What&#x27;s waas to fall over too? Thanks. Waas Green Square, My box. The answer No for our derivatives The durability for our original equation It&#x27;s gonna be ex can be smaller and 2/3 or x can be larger three House Because the bottom cannot be undefined Same goes for for, um no of time of eggs That&#x27;s must to be smaller than three No 2/3 war no more bigger three homes

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 27 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.
$g(x)=\sqrt{1+2 x}$

Answer

domain of the function $\left[-\frac{1}{2}, \infty\right)$
ddomain of the function $\left(-\frac{1}{2}, \infty\right)$

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domain of the function $\left[-\frac{1}{2}, \infty\right)$ddomain of the function $\left(-\frac{1}{2}, \infty\right)$

Biocalculus Calculus for the Life Sciences

Heather Z.

Kayleah T.

Samuel H.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartBiocalculus Calculus for the Life Sciences

Heather Z.

Kayleah T.

Samuel H.

Michael J.

domain of the function $\left[-\frac{1}{2}, \infty\right)$
ddomain of the function $\left(-\frac{1}{2}, \infty\right)$

James Stewart

Biocalculus Calculus for the Life Sciences