find-the-partial-derivative-of-the-dependent-variable-or-function-with-respect-to-each-of-the-ind-70

Question

Find the partial derivative of the dependent variable or function with respect to each of the independent variables.
$$u=\ln \frac{y^{2}}{x-y}+e^{-x}(\sin y-\cos 2 y)$$

Tyler Moulton · Accepted Answer

All right. We want to find the first partial derivatives of the function F of x Y is equal to Ellen, Y squared over x minus Y plus even the negative X times sine Y minus co signed to. Why? This question is challenging our understanding of differentiation as applied to multi variant functions. The first partial results were searching for our fx or DF dX and dY we use standard differentiation techniques. We learned in single variable calculus to find both, but for DF dx, which requires a constant. And for the FBI we treat X as a constant so that we can proceed to solve fx is three of Ellen Y squared over x minus Y with respect to x x minus Y over wide square times negative Y squared over x minus Y squared minus. Either negative X and y minus two Coast Costa Y. Or simplifying negative one over X y minus even em fx sign Y minus Costa Y. Then F Y is by a similar process of chain rule on an algorithm one over X minus Y plus two over Y, plus either negative X. To sign two Y plus coast Y as indicated the bottom.

Will Erickson · Answer

weather in this problem were asked to find all the second order partial derivatives of this function. So let&#x27;s begin with the first order ones. So the partial with respect to X Well, if you look in this first term, if we treat why, like a constant, you just get that the derivative. But why remains there and we&#x27;ll end up with two, uh, x times. Why, uh, the second term doesn&#x27;t even have an accident. So that will become a zero and 1/3 term. Unwise, a constant. And the derivative sign Exes co sign X. So we end up with to x y plus Why cool signed X. That&#x27;s not one of our answers, since it wants thes second order, partial derivatives. But we&#x27;re gonna have to use this so we&#x27;ll keep a note of it. And let&#x27;s take the partial with respect to why as well. So in this first term, now X is a constant. So the ex term will stay there. And why did the first just becomes one? We know that. So we get X scored from the first term. The second term, the derivative of co sign y, is minus sign Why and from the third term and exits just a constant So sign exes as well. So the derivative of Y times a constant will just be. They&#x27;re constant. And so there is G subway. All right, now let&#x27;s start the second order ones So we&#x27;ll keep all this by itself and start answering the questions. So let&#x27;s start with three sub x x. So we just look at G X right here. They get striven of now with respect to X so to X y. The derivative of that is just too Why and the derivative of y co sign X with respect to X. Why, it&#x27;s just a constant and the driven of a co sign X&#x27;s minus sign X So we get minus Why Sign X. So there&#x27;s one of our four final answers. Next up will take G sub y. Why so, same idea. Let&#x27;s look at G sub y and take its derivative with respect to why First term gives us nothing, since there&#x27;s no Y in there derivative of minus sign. Why so we keep our minus sign becomes co sign, and the third term doesn&#x27;t even have a wine it. So again, that&#x27;ll get zero So we just end up with minus co sign of why we&#x27;re almost there now. We just need G of ex wives. We need the two mixed partial drivers. Let&#x27;s look at Giza bex right here and take its derivative with respect to why. So the first term X is a constant wise variable, so to x Y and will give us the derivative to X and why Times Co. Sign X, and we&#x27;re treating wise the variable. So Coastline X is just a constant, and that&#x27;s all we&#x27;ll end up with their, um, Now we could invoke clear owes the&#x27;re, um at this point, since if we look at G right, all our functions, your continuous we have a problem. A meal on co sign and science. Everything&#x27;s continuous. I&#x27;m in multiplying. Continuous functions is a continuous function, so we could stop and just say this must be equal to G Y X. That would be acceptable. Or we could take five seconds and figure it out ourselves just for the extra practice. Let&#x27;s go ahead and do that. Let&#x27;s look at Jesus y and take its partial derivative with respect to x o. X squared gives us two X sign Wine doesn&#x27;t have an accident, so that zero and the derivative of Sine X is co sign X. So, as predicted, we get the exact same thing for the two mixed partials, and we are.

Will Erickson · Answer

mother in this problem were asked to find all these second order partial derivatives of this functions. Let&#x27;s begin with the first order partial derivatives W sub X. So treating excellent the variable here we can tell we&#x27;re going to need the product rule since we have one function of X times, another function of X. So buy the product will we&#x27;re going to need the first function X times the derivative of the second function. By derivative, we mean the partial derivative with respect to X of this second function. Okay, plus the derivative of the first function times the second function. That&#x27;s just the product rule that we know and love. Let&#x27;s go ahead and compute this. What&#x27;s the derivative of it? Sign of X squared y. Well, we&#x27;ll start with the chain rule here, since we have a sign of some function. So starting from the outside and the derivative of Sinus co sign, we keep the inner function here and we have to multiply by the derivative of this inter function again. When we say derivative, we mean partial derivative with respect to X. So since why is a constant the derivative of this inter function here is to x y that the wife stays there and derivative exporters to X to make sure you&#x27;re convinced. Event moving on the partial derivative of X with respect to X is just one, and the rest can stay there. So cleaning up her answer just a little bit the it to X squared. Why co sign next squared y plus sign X squared Y all right w sub y. It will be a little easier because this time X is treated as a constant, so we don&#x27;t need the product rule. That X is just a constant multiple. So that stays there, the derivative of Sign, He said his co sign, the Inter Function stays. Now we multiplied by the partial derivative of X squared y with respect to why which is just X squared. So you end up with X Cubed co sign X squared y all right, so we&#x27;ll circle those just so we know where to look these air first, our first order partial derivatives. Now time for the second. Let&#x27;s start with W sub x x. So we have another product rule to do so. Make sure you can follow along each step, feel fear to pause the video if you need to, at any point product tool for this 1st 1 just so we know where we are. We&#x27;re just taking care of this first term right now. So notice it&#x27;s foots into two parts. The two X squared y and the co sign of X Squared y. So how if we go, product rule tells us we need the first function to explain why times the derivative of the second with respect acts the derivative of coastline is negative. Sign of X squared y and the derivative of X squared y with respect to X is we said to X y all right, plus the derivative of the first function with respect to X. So the derivative of two X squared y is four x y Make sure you&#x27;re convinced of that times the second function co sign of export Why, All right, so we&#x27;re not done. It&#x27;s just that was the product rule of this first blue parte eso underlined this in blue. So we&#x27;re halfway there. We still have to add I&#x27;ll make it red, the derivative of sign of X squared y. This will be a little faster so the derivative of Sinus co sign. We keep the inter function now we multiplied by the derivative of X squared y with respect to X, which is to x y. Okay, so let&#x27;s clean this up. Um, so what do we have? Let&#x27;s see, this whole first term two times two is four. We do have a negative sign in there, so we should be getting negative for extra third y squared sign of X squared Wise all comes from the very first term, and now we have notice we have four x y coastline X squared y plus two x y Costa Next chord. Why those air actually like terms So the four on the two can add. We end up with six x y co sign of X squared y. That&#x27;s about as far as we can go. I think so. That is tthe e answer for wciv xx w. Some while the little quicker because we don&#x27;t need a product rule in that X to the Third Street is a constant again Derivative of coastline is negative. Sign. Let&#x27;s put the negative out front. It&#x27;s sign that inter function X squared y now times the derivative of X squared y with respect to why which is X squared. So we end up with minus X to the fifth sign of X squared. Y ok, getting there now. W sub X y this would really be a pain again, huh? To go Thio Tate to take w x and take the partial derivative with respect a wife, we would again need a product rule here. Um, so it&#x27;s clever to say, Wait a second. Look at all of our functions that we&#x27;ve found so far the original and all of our derivatives. They&#x27;re all continuous, all they ever polynomial ls and signs and co signs and all those air continuous And so we can invoke Claros. The&#x27;re, um there. I&#x27;m too from this section and assert that w x y actually equals W Y X. And in this case, it&#x27;s smarter to do it this way because, um, this should save us a little time, right? You will still need a product rule if we&#x27;re looking at, uh, taking the derivative of this down here with respect to X. But at least there&#x27;s only one term and not too. So it&#x27;ll save us a little bit of time So again we have a product to A We have a first function in a second function. We&#x27;re taking the derivative with respect to X. So the first function next to the third. Now we want times the derivative of the second part, the derivative of co sign is negative Sign uh, X squared y times the derivative of X squared y with respect to X, which is to X y Kim and the second half of the product rule Now times the derivative of the first part. That&#x27;s three x squared times a second function. Okay, and we can rewrite that down here. So let&#x27;s see what we&#x27;re going to have. The first term will be negative, and we&#x27;ll have minus two X to the fourth. It looks like exit the third times X. Yep, we do have a Y in there, and it&#x27;s a sign of X squared y, and in the second term is a simplified as it can get those three x squared co sign X squared. Why, all right, we&#x27;ll put a little boxer on that, and we&#x27;re done because that was both of our mixed partial derivatives equals this. So we have found all four. Hopefully, that helped

Jack Chen · Answer

for this purpose. We have a function if because Teoh t to exploit its my must sign of y square minus x 31 half times likely would have and they want to can put some pressure on securities. So we start a wave on the passion curative off with respect to x seven f sub X equals four them people for its my notes on my half its three minus one half times white you have had. So the sick Holder Panshir directive with respect to X will be 16 ft before x fast 1/4 a T minus three half times, likely last test and the Knicks order f sub X y We&#x27;ll be minus 10 or four x t minus one half times y to the n minus one half the patron narrative we&#x27;ve with respect. Why will be minus coastline y square comes to life minus one x 31 half times wider in minus one. So the sexual under pressure directive with respect why will be miners que close that wise, clear way need to use the product rule here because both terms uncontained Why I asked for why square times styled why square. Wonderful. Exactly. One have times y minus three and then a third of the posher through. Is that why? Why X will be Why not write X minus one? I&#x27;m delighted to mind this three because I&#x27;m the first to turn so need content. Why? So before you take one more curative with with respect to eggs, this two terms vanish.

Sriram Soundarrajan · Answer

today you&#x27;re going to sort probably number 36. He had a function is given with trust in extent play format, which is given but weren&#x27;t minus because do we do expressly but because the square expressway. So we have to differentiate with respect to eggs. It is given by zero minus my I say off through a window expressway in the bluff do cause in the express lane into my notes. I enough! I think it&#x27;s like, Okay, what do the bigger thing? It&#x27;s leg do enough The window expressly End of great miners saying off through its plus like that is it is a wonderful bulls eye in Digger costed up. It&#x27;s gonna join us. Sign. Did this one so visual? Becoming us sign toe? It&#x27;s less like And you the doctor, Did you agree with us? But right in the off. What a bigger being. Affleck Seattle minus. Why not say enough so explicitly in the bluff? True, because pigs less like in the minus sign It&#x27;s Leslie. We just given by we&#x27;ll say, do explicitly my left side so explicitly. Well, if you end by saying off you, that&#x27;s Leslie. Thank you

Linh Vu · Answer

in this question. Well, even the function next one to go to war Manas Course I after two x plus two y know, Plus course I square under thanks. Plus why? And here to do this question to find a first pressure The river there was watching the banks. Well, can&#x27;t this one with equal Jew Now the the X will be the variable. Why would be the constant? So they give them the wonder coaches. Oh, the real thing because I could yield a minus eyes on the other side Now off that you x plus two y by the general manager terms the review on the inside. So again, opportunity to and now Plus in this one win each of lined Uh huh power off us So have a chew course I&#x27;m the X plus y and by the general manager tams the rift in the course I could u minus size or have minus near. So I am the express quiet and you find a pressure there. If there was but the why noticed them there&#x27;s only can use their Smith&#x27;s room through in the X and going. So we need to industry in the X and y because off, uh, symmetry. So the person error of their ways Minnesota Why included to Sigh two x plus two y minus two cosa and express qy sy Express quiet.

Problem

Evaluate the indicated partial derivatives at the…

Problem 26 Easy Difficulty

Find the partial derivative of the dependent variable or function with respect to each of the independent variables.
$$u=\ln \frac{y^{2}}{x-y}+e^{-x}(\sin y-\cos 2 y)$$

Answer

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. Evans

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