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Where is y differentiable?y = sin(Irl) 0i = > 0olz < 0For all values of 2_Everywhere excepI 0...

Question

Where is y differentiable?y = sin(Irl) 0i = > 0olz < 0For all values of 2_Everywhere excepI 0

Where is y differentiable? y = sin(Irl) 0i = > 0 olz < 0 For all values of 2_ Everywhere excepI 0



Answers

If $f(x)=\frac{x^{2}-1}{x+2},$ where is $f$ not differentiable?

So we have our function here, X squared minus one over X Plus two, and we're being asked when it is not defensible. So let's observe the denominator here. When is dysfunction undefined when X plus two is equal to zero, so X is minus two. So if X is minus, two year function is undefined, so not defensible at X equals minus two. There's no continuities here. The your function would probably would increase toe infinity plus infinity indefinitely, so you cannot really draw a tangent there.

So first question were you in? The F X Y is equal to zero when experts lessen the line, which is lesson to X squared and one otherwise. And we want find three things personal in a show that the partial derivative of X and the partial derivative respect of why at Sears there exist. And then, lastly, we want to show that f it's not differential wool at that point. So first I want to find the partial derivative with respect X, And for this I am going to use the Lim definition. And so that tells me that this is going to equal to the limit as H approaches zero. Since I'm taking partial derivative respect of X, the plus h goes with the ex turn here. So when I have a church as my ax zero as my why, I see that this is obviously not going to hold a just not having a lesson zero and suddenly lesson, too. It's weird. So that's it means that this 1st 1 is going to be one when I have X, and why both equal to zero this again? It's not gonna hold because they're the same. So it's gonna be one also. So therefore, that first partial derivative is equal to zero. Said does exist now, taking the partial derivative with respect to why when did the same thing some limit as a purchaser of now f of 00 plus h stating the partial very respectable. Why so again Doing the same thing. I really know that on tickle the one. So now I have to see what happens when zero is accent H is good. Why? So this first part spent a whole h is going to be greater than zero. Most likely. However, Hey, the 2nd 1 is not going Tio not gonna hold. So this one's gonna equal one. Therefore you zero again. So we just showed the first part that partial drew Destructor X and Y and Z or exist. Now I want to show that the function is not differential at that point, though. So if this were needed to look back at the themes that give us nice seeing them for that says if f x Y is not continuous at zero, then it's not differential blood 00 I think it's the converse about them. So I want to shoot for a show that the function is not continues to 00 and I know it's not continuous of the limit at that point at the function does not exist. So I wanted to use my two pathway through room, picked two different paths, and so they come up with different limits. So first I want to go along. Why equals X squared? So basically choosing two different paths that will one ended but zero another end up at one. So if I choose along, why equals X squared limit as X Y goes to 00? But my function I see If Y is equal to X squared, then clearly it's not greater than X word. So that's going to end up equaling limit as X Y goes to 000 which is zero. Next, I'm going to go along. Why equals this time? I do want it to fall under here, someone to go ahead and put it at 1.5 x squared. And so then I get that the limit as X Y goes to 00 concussion has been away of my function. They're just going to go the limit. That is why this zero of one because it does fall under this case. Gonzo. No. Mix it up their asses up tough One ship in one. So I see both of these are not equal so that I know about the past. My serum that therefore so first therefore the limit as X Y goes to 00 X y does not exist. So therefore, Methanex, why is not continuous that 00 And so then by fear for f of X y is not defensible. That's it zero.

At X comma. Why he calling extensive widest seconded by X squared. For we went at common wise 1900 And the other one is with zero blowen X y equals zero euro So first we're showing that we should That primary exportable x zero comma zero and at partial y of zero comma is your axis But it's not different Triple zero comma zero. So when orphans the work you do? Uh, uh you know, to show his first mute No, the f prime of X and y time. So this is this is at X for X Connell. Why would your people to the partial in terms of export f x comma? Why? Which is feet cool too wide? The six minus X squared white is second These extremes Why does six divided by X squared plus wife Fourth, where we do have our show by X comma Why would you think partial? Why f x comma? Why Equalling you x third wide minus excess haired white in third minus two ex wives that fit by ex Coulier plus white before square that in our using the third dear um, three of you choose up that if the possibilities of ffx and f y of the function f x com A wire continues throughout the open region than F is confused at every corner. So we use the limit for using when burst with access people toe white The second the use wives is second comma. Why, as it goes to goes to zero Commons Europe for death of acts which is gonna be equal toe minus quarter. Then we do the limit. Uh, before that, we'd use instead of access one white a second. We've used X equaling minus white. Second, that which is a limit for Maine is white ist second come Why things goes Teoh 00 Why off x come a wide which is gonna be equal to the limit minus y squared times Why going Teoh zero commas your, um uh minus minus quietly eight Close. Why do they eat what you need divided by two times y to the fourth squared And by cancelling off the wise you'll get, you should be able to Yeah ah, one quarter since their bolt not the same f x comma, why is not deep occurring? So it wasn't different Trible It will not be since he's not defensible at, uh at zero comma zero, you will not be you

We want to find and be such that f is differential herbal everywhere. Let's differentiating Yeah, for the first function because I explain we differentiate, we get minus sign for the second function When we differentiate x plus b, we're just left with e If we were to grow up at crime, this is the Putin we're interested in. A supposed to zero minus sine x will look something like that here. And the constantly would just be somewhere here because at this point, for every time to be defensible, these two have to be the same. So, uh, X equals to zero You consider at prime zero, they have to be the same. So minus sign zero, it's equals two A. So a is zero Now plant the zero back into the function. We'll have cosine X expressed in zero, and we have be X squared equals zero. Now, at this point, X equals to zero. These two have to be the same as well. So consigned zero is equals, Toby. So V is one. Therefore, a zero in bes one


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