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(10 points) Find dy for 2cy +T sin y = 2r and use it to find equation of tangent at the dx point (1, 2...

Question

(10 points) Find dy for 2cy +T sin y = 2r and use it to find equation of tangent at the dx point (1, 2

(10 points) Find dy for 2cy +T sin y = 2r and use it to find equation of tangent at the dx point (1, 2



Answers

Find $ dy/dx $ by implicit differentiation.
$ tan (x - y) = \frac {y}{1 + x^2} $

So for this problem, we're gonna be doing implicit differentiation, which is a very useful technique when dealing with multiple variables in a function or wise and excess all throughout a function. So we have in this case is that because sign of X Y is equal to one plus the sign of wine. So we want to perform inputs, differentiation. And to do that, what we're going to dio is take the derivative this side and then take the derivative of this side. First, we'll take the derivative of the left side. Um, when we take the derivative left side, we remember that the derivative of cosine is negative sign. So what we'll have here is negative sign X y But then we have to do the chain role. So that's going to give us Times X. Uh, why prime? It's gonna be the product role. So x y prime plus why the x t x, which is just that which is why then we'll have that this is equal to the one will go away. Since it's the driven of a constant andan, all we'll be left with is the co sign of why and why prime So this is what Well, have right here and then we'll do next is we're going Thio Um we're going thio Add x sine x y to both sides and the reason why we have the negative x sine x y is because this right here is distributed. So, um, what will have in this case since this is all distributed is X It's a negative X sign fine times like crime. And then that's minus sign of X Y yeah, on Ben, we also want tohave. This is minus Why sign of ex wife? Just thio confirm that then what will have as a result is since this was all way, have to keep mine. This is all to be I distributed with So this is gonna be equal to the same stuff over here then our goal is to ultimately isolate the UAE prime. So we're going toe Add x sine x y y prime on both sides. So when we do that, we end up getting this right here. So it's positive announced in a negative. We can factor out the why prime on, we'll get a positive here. So be plus all of this and then lastly we can just divide this whole thing right here. When we do that, we end up getting this as our anthem for white crime. It's negative. Sign X y over coastline y plus X sine x Y, um, and that's our D Y DX and our final answer.

For this problem, we're gonna be doing inputs of differentiation, which is incredibly useful when we have excess and wise, um, in an equation or we have two functions being set equal to each other. So how it works is we're gonna take the derivative of both sides. We have the inverse tangent right here. The inverse tangent of X squared. Why, equal to X plus X y squared? I will take the derivative of both sides. Um, when we take the derivative of the left side, we're gonna have to use the chain rule. We know that what we'll get is, uh, going to end up being since the derivative of the inverse tangent is 1/1 plus U squared, for example, we'll end up getting is, in this case, 1/1 plus x squared. Why? But then we have to do the chain rule. So it's gonna be times on then everything in here. So it's gonna be X squared. This will be the product drawer here. So X squared kind of my prime. Plus why times two x, a chain right there. And then we take the dirt of this so we'll get one because the derivative of X. Just one eso We'll have one plus and then off to do change rules. That will be X times two. Why? Plus why squared plus two i times why prime plus y squared thought that we end up getting here. As a result, we can multiply everything through right here. So what we end up having as result, is going to be X squared over this portion down here. And then, um, that will be in this case since we had to square this portion. Since we have to do you squared? This was squared right here. So what? This is actually gonna be X to the fourth. Why squared? And then we'll have minus two X. Why, Onda? Another thing we have to keep in mind is that in having the minus two x y, this is going to be all times de y dx, which we can write his wide crime. I will be equal to one plus y squared minus and then two x y over one plus x to the force wide squared. So the reason we get this is because we know that, um, we can combine these on dwhite. We have the negative two x y This is coming from our distribution right here, the white Times to access to X y. And we know that it will end up becoming, um, minus that value when we subtract it because we have two. X y here is well than what we can dio. To simplify things further is we're going to divide both sides by this portion right here. So what will end up getting as a result when we divide all this is we're going to have that why prime is equal to one plus X to the fourth wide squared plus why squared US X to the fourth. Why to the fourth minus two x y, and that is all going to be divided by X squared minus two x y minus two x to the fifth y Cube. And that will be our final answer for why Prime

If this problem were given any question in the West, who's the petition to find? Uh, there is a suspect. So we're gonna take their old suspect X. So we're gonna use protocol for Derek about the personal left inside and political Or do it the second German left inside. So let's start taking their He had Sandra Boy plus X times. So sign of why he liked gets that is a little first term plus, um Why? Times Co. Sign X plus Sign of x times. You are the eggs that is equal to zero since one. It's just constant. Celeste were old terms with you I v x or Left Inside you have extension centre for plus sign X and let's write all the terms. We got the idea. Exxon. Leftist. Right inside. So it that is negative. Why co sign of explosive? Why promise? We don't see that they were told by with Spector. Excuse. Why the ecstasy? Good thing, too. Oh, why her sand x plus sign of why Divided by X times goes on. Employ plus Cy

He's probably given any questions that were asked to use interest depreciation to find a video file. Respect, excellence. I tear it off all the terms with respect effects. Let's start from the German left inside. We're gonna support hope. So we have Sign Ex Turns. Why, plus co signed X times you want Begin Since why's function X on the left on the right. Inside we have two X plus two wine again you. And yet, since Y is a function extra less group all the terms with DVDs one side. Unless group over terms, you got a few ideas on your side. Let's write this one as you want the eggs off to while my disco side effects. Is it going to, uh, we have? Well, that would be negative. Selects since they were two courtside inspector sign. So we have negative. All sign next times. Why minus two X. So basically, I took this Theo idea next time and right here into that two extra men riding on the back. So we haven't divine the answer for everything to be negative off. Why time sign Next plus two x divided by two. Why minus co signer. Thanks or that is equal to white time. Sign of X. You know, I don't like to x divided by co sign X minus two. What?


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