Question
Dz 9. (5 points) If x3 sin(2) + (24 1)y? = 2 find dy
dz 9. (5 points) If x3 sin(2) + (24 1)y? = 2 find dy
Answers
Evaluate dy/dx at the given points. $$\left(x y-y^{2}\right)^{3 / 2}=5 y^{2}+3 ; \quad(5,1)$$
Okay and this problem we're gonna be using implicit differentiation to find dy dx and we're going to evaluate it at the point to negative one. So my first step here is we're just gonna do implicit differentiation. So for this first thing we have dy the derivative of D Y is just gonna be too but then we have to put the dy dx And then we have five Since five is a constant number. The derivative of five is just gonna be zero. Then the derivative of negative X squared is negative two X. Then we move on to negative y cubed. So the derivative of negative y cubed is negative three, Y is squared dy dx. And then the derivative of zero is 0. So we want to get do Y dx completely by itself. So I'm gonna get rid of this negative two X. By adding it to both sides. Mhm. And then I'm going to factor out a dy dx. So I'm left with two three Y squared equals two X. So my last step to solve for Dubai dx would just be to divide by two minus three Y squared. Mhm. So those would cancel out and grd X equals two X over two minus three Y squared. So now we need to evaluate dy dx at the point to negative one. Mhm. Which means we just substituted to for an X. And the negative one for why? So this is really gonna equal to times two Divided by 2 -3 Times -1 Squared. So we just need to simplify this And we get dy DX at 2 -1 that's going to equal for over two minus negative one squared is just one. So two minus three Which is just negative one. Which means dy dx at this point would just be equal to negative four.
In this problem. We're doing any question and were asked to use implicit differentiation by very why just X. So we're gonna stay there with your old term to expect extent. So let's start from the first term we have or execute. It was for the second door getting portable. So we have two extremes. Well, spread plus explore eight times to wine time since Y is a function of X feels happy I d s plus three U S grade. Again wise function X will be happy by the ex again and that is a zero sense. Why is constant? So let's cripple terms with Levi the ex left inside. So we have some ideas. Oh, thanks to expert wine plus three Westbourne that is equal to then negative to x Oh, wife's Craig waas two squared their promise We don't see that very hopeful I respect eggs physical through negative talk to x Times y skirt was to experience the wild by two x squared plus three wire. Uh, sorry to ex Craig. Why? Plus three one square
You know this problem? We've been given the following curve and we would like to find dy dx by implicit differentiation. The first thing we do want to do here though is rewrite this and so it's right to this. Two X. To the one half. Well it's wider. The one have mm Is equal to three. And then now we can take out a really good from this. The derivative two extra 1/2 Is X. to the negative one half. Why do the one half is 1/2 Either the negative one half. Why prime In the derivative of 30? If we subtract tax and they have one half over. This gives us one half. Why do the name of 1/2? Why prime Is equal to 95? That's the one here. Right? Multiply both sides by two. It's going to get rid of that 1/2 And then divide by Y. to the negative one half. And this gives us why prime is negative two X. The native one half Over wide of the -1 house. Now those negative experiences mean flip where they are in the fraction. And so this tells us why prime does negative to route. Why overruled X.
Well that's probably been given four cosine x. Sign of why Is equal to one. And we would like to find the derivative of this. Use an implicit differentiation. Now in order to do this we will have to make use of both the chain rule and the product. Mm. Now the derivative Of four Cosine X. is negative. four cynics times the second there. So that's times the sine of wife plus the derivative of the sign of why is a co sign of why times Y. Prime. Mhm. Times are 1st of Times four co signing. And then the derivative of one on the other side is your I'm gonna take the negative for arsenic sign Y over to the other side. And so this tells me why Prime times for co Synnex chosen Y. is equal to four Synnex sign wine. And then all divided signed by four coastline, expose on wine. The legislation with the white prime there on the left. Yeah. Yeah on the right. The force cancel. Cinemax over Cosine of X. Is the tangent of X. And similarly the sign of Y. Over the coastline. Boy is the tangent line