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Implicit differentiation to find y'.cos (xy )=x"_yy...

Question

Implicit differentiation to find y'.cos (xy )=x"_yy

implicit differentiation to find y'. cos (xy )=x"_y y



Answers

Use implicit differentiation to find$\frac{d y}{d x}.$ $$x+y=\cos y$$

Let's start by finding the derivative on both sides. So we're gonna take DDX of X plus y. This is gonna be equal to d d x of co sign of why So on this side here, DD eight x of acts is just gonna be one and then plus DDX of why is gonna be d y dx it's gonna be able to So the driver of this well, first we start with co sign derivative coastline is a negative sign. So negative sign of why. And then we're gonna multiply all that by derivative of why which is D Y t X because of chain rule. Next, we're gonna move this over to this side all of our d y DX terms to this side. And then we're gonna move this one over to the other side because it doesn't have a d y t X. So we have d Y d X plus sign. Why times divide E X is equal to negative one. Scoot this over. And then now we're gonna factor out a d y d x from both of these terms. So it's gonna be times one plus sign why? And this could be able to negative one. Lastly, we just divide by the one plus sign. Why so D Y d x? Yeah, When we read it as t Y the X this is gonna be able to On the bottom, we have one plus sign of why and on top We have negative one. Just our final answer.

We have CoSine of X y squared equals. Why now We're going to go ahead and implicitly differentiate with respect to X So this gives us a negative sign. X y squared. So we've just taken the derivative of the cosign part. Now we still need to multiply by the derivative of the inside of the cosine function. So the derivative of X times y squared That's a product rule that we need to do. So the derivative of X is one. So we have y squared plus x times the derivative of y squared two y y prime. And on the right hand side, we have flight crime. Okay, let's go ahead and rearrange. Uh, rather let me just go ahead and boil this out. X y squared Why squared minus sign X y squared times two x y Why prime is equal to why prime Now let's go ahead and group all the UAE prime terms on the right hand side. So we have minus sign Ex wise squared. Why squared is equal to sign X y squared times two x y plus one times Why prime And now we're going to go ahead and divide through by sign of X y squared two x y plus one. This is a squared here one plus two x y sine x y squared and that's it.

For this problem here. We're going to take the derivative with respect to X on both sides. So doing that derivative have signed X design lips cause I know why. And then the sequel to D the x of sine X plus cosigned. Why so to take the derivative with respect to X here, we're going to need to use product rule. So we're gonna do first times derivative second, so sign of X and then times the derivative of this here, so derivative of co sign is negative. Sign why? And then we need to multiply that by the derivative of why which is gonna be D y t x Next we're going to second times derivative first so co sign of why and then times River sign next is just gonna be cosigned X here. So now on the right hand side the derivative of sine X here is just cause I necks and then derivative of co sign y It's gonna be negative Sign why And then we're going to need to multiply by d y t x here because of chain role. So then now we're gonna move all of our d y t X terms over to one side. So move this one over here and then our non b y t x terms to the other side over here. So this becomes a positive sign. Why so d y d x sign? Why? And then this is a negative term. So write it as a then minus and then we have sine x times sign why? And then this is gonna be able to close I in X, then minus cosigned X and then cosigned. Why now I'm going to, um there should be a Z y t X term out front here. By the way, I'm going to factor out a d y t x and a sign why as well so do wide the X sign. Why? And then we're gonna have one minus sine x and then on the right hand side. I'm also going to factor out a co sign X. So the Cisco sign next times one minus cosigned. Why now? I'm gonna divide both sides by this whole term here, so that leaves me with the wine. The X it's gonna be equal to it will have cosigned X. Um, I'm gonna write it as one minus cosigned Why? Times cause i X all over one minus sign. Why or sine X and then sign. Why? So this is our final answer here.

Mhm. In this problem we wish to find dy dx via implicit differentiation for the equation Y equals X times coastline. Why this question is challenging understanding of differentiation in particular. As a question suggested to challenge your knowledge, implicit differentiation for which you must have the following knowledge and understanding. First we must assume that why is the differential function for X as an example Db X Y n equals N Y n minus one dy dx. We must differentiate both sides of the equation. Inspector X. Making the assumption from above and then solve the resulting differentiated equation for dy dx. So this information in hand we all we need to solve. Let's proceed to differentiate both sides. D. D. X. Y equals D. D. X. X. Cause Y left hand side of the power will become dy dx. This right hand side becomes by trigger metric power roll and product will coast y minus X and Y. Dy dx. Thus pushing the dy dx terms the left and dividing by the fact that multiply gives dy dx equals cosine, Y over one plus X and Y.


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