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Findby implicit differentiation.X +yf =20dy dx (Type an exact answer:)-r3icEnter your answer In the angwvor box:...

Question

Findby implicit differentiation.X +yf =20dy dx (Type an exact answer:)-r3icEnter your answer In the angwvor box:

Find by implicit differentiation. X +yf =20 dy dx (Type an exact answer:) -r3ic Enter your answer In the angwvor box:



Answers

Find $ dy/dx $ by implicit differentiation.

$ x^4 + x^2y^2 + y^3 = 5 $

Yeah. It's probably want to find the derivative implicitly of X squared plus Xy my wife's bird because they're beautiful. No. And take remember able to hear the derivative of X squared is two X. For the derivative of X. Times why when you use implicit differentiation and the product. So the derivative of the first is one times a second plus the derivative wise by prime time is the first, then the derivative of Y squared as to why times why prime And the derivative of 40. Now let's take the two x. And the Y over to the other side. And on the left hand side will factor out a Y. Prime. And so that's why prime times X minus two Y equals On the other side -2 X -2 Y. So now divided by X -2 Y. Okay. Is this why prime Is -2 X -2. Y Over X -2. Y.

Yeah that's probably have been given the curve X to the 4th times X plus Y equals. Why do the second Times three X -Y. And we would like to find the derivative of this curve. Use an implicit differentiation. Now before I take the derivative I'm going to distribute in on both sides just to make our lives a little easier. So this becomes actual 5th plus That's the 4th y. And on the right hand side distributed in the Y squared gives us three X Y squared minus. Why cute. And so now let's take our driven Really? That's the 5th. This five and so forth. Yeah the derivative that's the fourth Y needs the product rule. And so we have four X cubed Y plus why prime and so forth. On the other side three xy squared also needs the product. And so this is three Y squared Plus six X. Y. Why prime rid of like cuba's minus three Y squared what crime now we need to solve for R. Y. Prime. Sure. And so I'm going to take the Y. Primex the 4th over to the other side. I'll take the three Y squared. Since it doesn't have a white prime minute over to the left. And so we have five. That's the fourth plus four x cubed y -3 Y squared there on the left equals. And then I'm gonna pull why prime out on the right? My prime times six X y minus three Y squared minus actual fourth over there on the right I'm gonna divide each side by that. Six xy minus three Y squared minus X. The fourth. It's just going to leave me with that Y prime there on the right and I don't know, I have to have five and so forth, Plus four x cubed Y -3. Weisberg Over six, XY minus three Y squared-. That's the 4th. Mhm.

This problem, We're gonna be doing something What's known as implicit differentiation, which is very useful if we have multiple variables like y and X and a function. And we need to take the derivative of why with respect to X So we're given X squared minus four x y plus y squared equals four And what we do is we take the derivative of both sides. Now it's obvious that if you take the derivative of this site since the constant, it'll be zero eso. Now we want to focus on the derivative of this side right here. DDX So we know that it will be the same thing is taking the derivative this minus the derivative this plus the derivative of this. So what will end up getting as a result? Um, it's fairly clear here that what we'll get is, uh, two X, that's the derivative. Then we'll end up getting minus for why, But then, since we have, um, that right there, we'll have minus four. Why? And then Plus, uh, since we have the wire right here, what will end up having as a result is going to be a why prime, So multiply this by white crime. Then in addition, will have plus y squared. The driven will be two y I'm white prime So now what we have is two X minus four y minus for X Why prime? Because we want Thio ultimately have the change well done. So it will be minus four x times y prime plus two y times. Why prime? And like we said, that will be equal to zero. Then we can factor out the white prime here. So we do that. We end up getting a two x minus four y plus a negative for X plus two y with the UAE prime factored out equal to zero. Then we move over the two x and four. Why? So when we do that, we'll end up getting right here. But we have to switch the signs. So the plus for why minus two X Then we want to divide everything by this right here. So I'll take that great and divide it. We know that the queues can cancel out, so it will be a well cancel it out with a negative too. So this will be X minus two by over Xu acts minus wine. That will be our final answer for why Prime

This probably won't use implicit differentiation to find a group of a two X. Youtube plus X squared Y minus X. Y cubed Is equal to two. And so in order to find a derivative here where you don't use implicit differentiation. Now the groove to execute is six X squared. Now for the driver of expert why we need to use the product rule. The derivative of X squared is to actually have two X. Times. Why plus the derivative of Y. Is why crime times X. Word. And then now I need the derivative of X. Y cube. The derivative of X. Is one. Mhm. And the drought of white cube is three Y squared by prime. And then we have times yet he put the white cube. Keep would have white crimes in them on the same side. And then pull why prime out? This is why prime times X squared -3 X. Y. Squared. I'm sorry Southside zero. The directive of 20. Yeah I know it's zero and then take everything else over to the other side. So it'll change signs so it's negative six X. Word minus two. Xy plus. Why cute? Mhm. And then divide everything by X squared -3 X. Watchword. And so this gives us why prime does negative six X squared minus two. Xy. Close Y cubed over. That's weird minus three X Y squared. That's what you're in.


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