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Question

Solve the given problems.
Find the derivative $\frac{d y}{d x}$ of the implicit function $x \cos 2 y+\sin x \cos y=1$

Tyler Moulton · Accepted Answer

we want to find Dy dx for the function X. Coast two Y plus sign X. Cosign Y equals one to do so we rely on differentiation shortcuts we picked up through a single variable calculus, particularly Rule +03 listed here on Electoral zero is D V. X. In Mexico&#x27;s Cossacks. GDX Coast Mexico&#x27;s native syntax rules one through three. Other power rule product rule in chain rule respectively. Were also going to be relying on rule three in the sense that because both sides of this equation include Y. Or rather this is not simply a function of why we&#x27;re gonna have to reform implicit differentiation. So we have an implicit differentiation DDX exposed to Y plus sign. Mexico&#x27;s Y equals C. D. X. One. The right hand side of course just become zero and are conservative becomes coast to y minus two. X. And Y. Dy dx This from product in the first term plus coast, ex coast Y minus sine X. And Y. Dy dx. Product in the second term equal zero. Thus solving for DY DX gives you I. D. X equals Coast two Y plus Coast. Excuse why over two X into Y, person, X and Y.

Karl Schaefer · Answer

in this problem, we&#x27;re asked to find e y d x and D squared wide DX squared given the equation sign of why equals x co Sign of y minus two. So it was for a start, my friend in the first derivative. So we need to take stir the derivative both sides. So on the left hand side, we have sign of lives. We kind of want to think of. This is like a chain rule. So the sign is our outer function and that&#x27;s why is our inter function. So when we do, the chain would take the derivative of the outer function driven Science Co sign plugging the inter function that multiply by the derivative of the inter function. So co sign of why the y d X now on the other side. We need to do it a product rule here. So it&#x27;s the drift of the first times the second plus the first times the drift of the second, so to differentiate coastline why wouldn&#x27;t do the same thing? Is a different shade sign of why we have a chain rule outer and inner derivative of co sign is minus sign ten times the derivative the inside do I d. X and then the minus two goes away because the dreaded of two is zero. Now that rearrange this to get all the d Y d X is on the one side So we&#x27;re just going toe Add this term over to the other side coz Why do i d x plus x sign? Why d y d X equals cosign? Why your factory do i d x out of the left hand side And now we just need to divide by this term Coastline y plus X sign wide to get our final expression D Y d X equals coast. Why over Coast? Why plus X sign? Why? Here&#x27;s the first derivative. Now let&#x27;s go to a new page and start thinking about the second derivative. So first we&#x27;re going to write the first derivative again. D Y T X is equal to coast. Why over coz why plus X I&#x27;m y now we want to find the second derivative, so we just need to take the derivative of the first derivative. And now we see that we have a question here. So we need apply the quotient rule So the second derivative apply the crow shit rule So low d hi Minus Hi, Di Lo. So the bottom post five plus X timeline times the derivative of the top. So on the last page, we already took the Drew two of co sign Why, it&#x27;s minus sign. Why do I d x minus the top times the derivative of the bottom So is taken now the truth of the bottom drawer of cosign Why is minus sign? Why do I d x plus x times sign of why this again We did take the druid of this on the last page. It&#x27;s s a sign of why now we need to differentiate. Sign of why So it&#x27;s co sign. Why do I d x by the same process we did on on the last page, all divided by the denominator spread. So this looks pretty bad and this is this is not quite a final answer because the problem asked us to put our answer in terms of wise and exes. But we have d Y d X is floating around now we do have an expression for D Y d X that we can plug in tow all of these terms, but There&#x27;s just going to be really big and disgusting, and we can actually do a little bit of simplification first. So what I want to do is I want to expand all of these terms on the numerator and kind of see what happens. So first on the left hand side, we have minus sign. Why Coast? Why do I d x minus x times sine squared? Why D Y d x so I just multiplied. Distributed this out here Now in this silence distribute the minus co sign so minus coast. Why time minus sign? Allies plus coast. Why sign? Why D y d x minus coz Why sign why minus x Cho squared y do y d axe and then all over this denominator But now we see that we could do some simplification. So, for example, here&#x27;s a minus sign Why coastwide E Why d X plus sign? Why coastwide u i d X. And the other simplification that we see is in this term and this term we can factor out a minus x d Y d axe. So let&#x27;s do that. We&#x27;ve cancelled these read terms and now when we combine these blue terms, we get minus x d y t ax times sine squared wide plus coast squared y The other term left in the numerator is minus. Coach. Why sign? Why? Divided by coastwide plus X sign, Why all squared? But now we can actually do this Really nice simplification that we know sine squared plus co sign squared is just equal toe one for any value of why so we can simplify this and just get a one as a final step. Let&#x27;s plug in our expression for D Y d X right here. And so our final expression. Our final answer is minus x times D Y d x, which was coast. Why divided by coast? Why, plus X sign. Why, then the science fair Priskos Time&#x27;s Pretty Girls one and then minus coast. Why sign Why multiplied together all divided by this denominator Cose. Why, oops made a stray mark here Coast y plus X, I&#x27;m signed. Why squared? So here&#x27;s our final answer in terms of X and Y for the second derivative D squared y over DX squared

Will Cadelina · Answer

to solve this equation for D Y over DX or the derivative of why we&#x27;re going to start by differential on both sides of the equation in respect to X. Okay, we&#x27;re starting with less side of the equation. The road of of co sign X Y is negative sine x y. Now we need to find the truth of the inter function because this is a chain rule I mean, funded room X y and multiplied by this and the driver of that is going to be, um, the root of X times. Why, though the room exit just one times Why just why? Plus, uh, x times that room. Why? Which is also one. But since we&#x27;re deriving in respect to X when he&#x27;s multiple, my wife prime or the rhythm of X, which is often called, is equal to the room of one is zero plus, uh, and the droop of sign of why is going to be co sign a Why again, Times y prime? Because we&#x27;re deriving in respect to X. Okay, now all we have left to do is just try to solve for y prime these rallies right here. So what, we&#x27;re gonna do is first multiply and distribute out our equation. Okay, Now that we&#x27;re here, uh, we&#x27;re going to subtract both sides of the equation by why Prime Coast on y to get this function. We&#x27;re here to the other side. We&#x27;re gonna add sign of X y and why this is We&#x27;re also going to factor out these values for their wife prime value with negative sine x y, which is just to rear. Inform Booth minus CO. Sign of why Which is gonna be equal to why. Sign of X Y. Now we&#x27;re going to divide both sides of the equation by negative X sine x y minus co Sign of why to isolate the wide prime variable. And this is your answer. You should interpret this. Why prime or the derivative y or dy y over d X is equal to why Sign of X y all over negative X sine x y minus co sign of why

Will Cadelina · Answer

to find d Y over DX or the derivative of y of this function, we&#x27;re going to start by differential on both sides of the equation in respect the X. Okay, let&#x27;s start with the left side. This is gonna be a product rule for why co sign X. We&#x27;ll start with the truth of the first function, which is why the derivative of Y is equal to one. But since we&#x27;re differentiating in respect to exodus to be multiplied, I wiped prime, which is also known at the root of why it&#x27;s also times cause the necks, they&#x27;re gonna add that to why time The derivative of CO sign X, which is negative sign of X No one would, where the fruit of X with just one so stays the same. So that&#x27;s the left side announced to the right side. The derivative of one is zero. So is matter. And now we have a sign of X y. And the truth of that is gonna be co sign X y, which will multiply that by the product rule. For the true love of X y. The root of X Y is gonna be ah drew of X one times why plus x times the truth about why which is also one. But we multiply it by white prime because we&#x27;re desperation in respect to X. And now we have a little stuff out. Let&#x27;s distribute all of our equations out first. Okay, Now we want to do is try to isolate for this. Why Prime variables and what we&#x27;re gonna do is get both of them to the same time equation so we can factor out. So what I&#x27;m gonna do is subtract x y primeco sine x y from both sides of the equation to get it over here and then also add, Why sign extra besides the equation to get this function to the other side? When we do that, we can also factor out prime variable while we&#x27;re doing it. Idiot Wife time Costa in X minus X close sine x y Okay. And now for the other side is going to be Why co sign X y plus why Sign of X And now what we want to do is divide both sides by co sign X minus X co sign of X Y Teoh, Isolate this. Why crime? And when we do that we can see our answer for why prime or the derivative of y or do idea Ah, the answer is equal to Why Why plus y sine x all over coastline of X minus x co sign X Y.

Carson Merrill · Answer

both probably want to find the second derivative using implicit differentiation. So we&#x27;re given in this case is sign. Why puts coastline ax equals one and the main reason we use inputs a differentiation is if we ever have functions in which the X and the Y are together. And there&#x27;s not just one derivative with sacred to take the derivative of both sides because we&#x27;re given an equation. Uh, if we&#x27;re taking the derivative of an expression, that&#x27;s different. But since we have this equation here will take the derivative both sides. When you take the derivative of the right side, it&#x27;s just gonna be zero, since one is constant. The left side, however, what will end up having is the sign y the derivative that would be co sign y. But it&#x27;ll be time to wipe crimes and sort of taking the derivative with respect to X. And then it will just be minus sine X. Since we&#x27;re taking the derivative with respect to X, so then what we&#x27;ll have is waas. Add the sign next to this side, divide coastline y over, and what will end up getting as a result is that why prime is equal Thio sign of acts over coastline of wine. Then we want to find why Double prime. So we&#x27;ll take the derivative of that right there. When we do that, we see that we will use the quotient rule. So why double crime is going to be equal to the low times the derivative of the high. So the denominator times the derivative so coastline Why times the derivative of the numerator, The derivative of the numerator is, uh, going to be the coastline of acts. We get coastline next time coats on why minus the sign of acts. I&#x27;m the derivative of the denominator. The derivative of the nominator is going to end up giving us, uh, this will be a negative sign y times. Why prime? And then all of this is going to be over the denominator squared. So it&#x27;ll be over cosine squared. Why, um then when we substitute all this in what we see is that we have coastline Next times coastline y minus sine x. These negatives will cancel right here, so we&#x27;ll just get plus sine x times sign why? And then we know that this right here is equal to the sign of acts over the coastline of y. So that would just make this cubed right here. And it&#x27;ll make the sign of ax squared so we&#x27;ll just do this. This will be our final answer for, um what we have. The only other thing that we would have to do is make sure that the cosine of why squared as well, because we need Thio. Multiply everything through. This would be our final answer for the second directive.

Will Cadelina · Answer

to find the derivative of X of this equation. Start by deriving both sides in respect to X. Okay, but left side, this is gonna be a product rule first. That&#x27;s going to be the root of Of why? Which is one times since its in respect to X. Why prime or are the root of of why so just why prime plus ah, which is gonna be why this is multiplied by coastline of X plus why times the druid of CO sign X, which is just negative sine x okay. And for the other side of our equation, it&#x27;s going to be the truth of of X squared, which is two x plus the truth of why squared to Why times the truth. Why my prime? Okay, now I&#x27;m going to distribute out all of my parentheses. Okay, Now that everything distributed out, I want to ah isolate the why variable. So the wife prime variable. Sorry, eso I&#x27;m going to move this one over the other side so I can, uh, isolated with the other wide prime and then also move this expression over the other side so I can cause it&#x27;s not times why prime so when a user. Okay, okay. Now I&#x27;m going to isolate. Ah, Why Prime factor it out. Okay. I&#x27;m gonna divide both sides by co sign X minus two y to isolate the white prime variable. In truth, why? And that could leave us with Why? Sign of X plus two X all over co sign X minus two y. And that&#x27;s your answer for the derivative of why of this equation.

Problem

Solve the given problems. Show that $\frac{d^{4} …

Problem 44 Easy Difficulty

Solve the given problems.
Find the derivative $\frac{d y}{d x}$ of the implicit function $x \cos 2 y+\sin x \cos y=1$

Answer

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