evaluate-the-indicated-partial-derivatives-at-the-given-points-zx2-cos-4-yleftfracpartial-zpartial-3

Question

Evaluate the indicated partial derivatives at the given points.
$$z=x^{2} \cos 4 y,\left.\frac{\partial z}{\partial y}\right|_{\left(2, \frac{\pi}{2}, 4\right)}$$

Sriram Soundarrajan · Accepted Answer

hello doing today? We&#x27;re going to start problem about tech T. What she d given function is that because crude off X squared plus four y square so knows that by the exit one call Mark because like by X Square, plus full by square indu elex So which will be getting us going to? But they will play to root. One plus would do Bruce quit, which is the goto wondered. But both one blessed 16 by group started similarly. Those there by no like that one cold march with physicals. Climate by X squared plus full. Let&#x27;s grab in tow. Eight. Right, which is because for why doing it by good stuff they prefer full life quit, which is because alright, Google, they were by We&#x27;re tough work less 42. It&#x27;s quite a logistical $8 but Group 30

Ashley Boni · Answer

So we have a function of X, y and Z, and it&#x27;s equal to the square root of sine squared. X plus finds work. Why less science were busy and we&#x27;re interested. And the partial derivative. Ah, with the respect Izzie evaluated at the 0.0 fire before. So first, let&#x27;s find our partial derivative s o The square root is the same thing that&#x27;s having a 1/2 power. So when we take the derivative with respect izzie, that 1/2 comes out front and then we keep what&#x27;s inside. So science were X plus signs. Word Why last time scored z No subtract one from the power So we have negative 1/2 and then change rule tells us to multiply by the partial derivative of the inside. Plus I sine squared. Why? What sine squared, Susie? So we&#x27;re treating X&#x27;s and y&#x27;s constants. So these guys are just going to be zero. So we need to take the derivative of sine squared psi. So when we take the derivative of sine squared, we end up with two sign co sign so to sign Z co sign. See? So now this two cancels with this 1/2 on, We end up with sign times co sign over this square root of sine squared X plus I&#x27;m squared Why plus sine squared c So now we&#x27;re just gonna plug in our 0.0 pi over four. So fire before our Z coordinate So we have sign high over four times co sign pi over four over square root of signs Square zero, plus sign square zero plus sign squared high over for So we know that sign of zero is zero so we can get rid of these two terms. Ah, sign of pi Over four is one over the square root of two and co signed a fire afford also one over square to two and on the bottom we have square root of sine squared So it&#x27;s just signed a pi over four again, which is one over square root of two So we can cancel one of these groups are too out and one over square to choose our final answer

Carson Merrill · Answer

partial derivatives are extremely helpful because when we&#x27;re in calculus and we want to optimize their minimize, we use derivatives. And oftentimes what we have is functions of multiple variables. Eso Therefore, we want to have partial derivatives to take the derivative of the different variables. So that&#x27;s what&#x27;s going to be the case for this problem here and were given a function. And we want to take two different partial derivatives that were given. The function that we have is the function f of X y Z, and it&#x27;s defined by the co sign of four X plus three y class Cuzzi. And what we want to find is first, we want to find f X y z, so to find f x y z. What we&#x27;re going to dio is first the partial derivative with respect to X um, and what we end up getting is a negative for sign four X plus three wide plus to Z. That makes sense so far. So then when we take F x y, then we end up getting a negative 12 co sign of four x plus three y plus two Z, and then lastly, we want to take the partial derivative of that with respect to Z, so we&#x27;ll end up getting as a result is 24 sign, um, 24th sign time or 24th sign of four X plots? Three. Why us to Z? And that&#x27;s what we&#x27;ll end up getting. Then we want to calculate, uh, the partial derivative of y ZZ. So if we look at, um, the partial derivative of F let&#x27;s respect why what we end up getting is a negative three sign, and we could just copy and paste this because we&#x27;re gonna keep reusing it. So that&#x27;s right here, then, um, f with the partial derivative Y Z will get a negative six code sign. Um, with this whole thing and then lastly, f partial derivative wise easy is going to be equal to taking that derivative of busy Variable will get a too negative, too, because it&#x27;s co sign. So it&#x27;ll go to negative sign. So the negatives cancel out and we&#x27;ll be left with 12 sign of this portion again. And those would be our partial derivatives. I&#x27;m right here and also right here

Linh Vu · Answer

So first computer f on, uh, accident. So why we meeting Constance? Which again? The three far expelling dream. Why, yes. Why, Mom? Honest Jew And asked why which? Is that the recipe? Really? Why? Action expert for our class. Uh, it becomes a mine. Us now minus Jew. Thanks. And why our monastery?

Linh Vu · Answer

in the course of the train. Thanks. Why? C e cogent for X minus two y minus. Juicy hungered three y minus six X minus three. And see, he had noticed that. And we trying on just simply find everything here. We can factor that do on site. And so have the X. Ah, yeah. Uh, minus know why Minus dizzy 90. Writing by work that conflicted a ministry on sign. Inside we have the x minus, uh, why and then bless Z. So we cannot cancel this one, so this will be fine. We shouldn&#x27;t do that. So can&#x27;t came the same for me, huh? So the first special here, everything was went to the x X will be the variable and the others Verbal will be the constant. So we need your blender caution room here three y minus six x ministry z ballot You down the river them with chemical too far. So we&#x27;re gonna fall stems So we have four times the denominator equity trail. Why honest 24 thanks. And now minus 12 easy. And now the riveting But Tom in culture fall. You know, they give them the bottom ical to minus six so no place six time with the numerator includes 24 x finest trail Why and now minus 12 c So it was going to find everything. We should get you some counseling with this one There were concerned with the wine There&#x27;s a can dizzy and then we get equal to minus 20 phones. Easy now True. I&#x27;m honest. Thinks X ministries e power to similarly the partially river, the group&#x27;s bachelor Why we should get now Women victory Why minus six Thanks My mystery Z about to and now new Everything on the numerator include U minus two times going to denominator temperature of minus six y plus 12 Thanks on now plus six C now minus the riveting that they nominate the Ottomans at three. So you have a minus three times with the top. So money trail Thanks plus six quiet and now plus six c. So let&#x27;s see if we can conserve something here. It&#x27;s when with this one is with this Don&#x27;t have the trailers e off the tree y minus six x minus three z. But what you now the partial derivative was watching to see if he called you. You have a dream woman A six. Thanks. Ministries square again. They give them the Tom good human minus two times when the denominator equal to minus six. Quiet plus trail. Thanks. And then blessed six c And now they really don&#x27;t know Denominator equal to minus three. We&#x27;re never gonna three toe have a blessed three Blessed trail. Thanks Now and now minus 61 And we get a minus six z. See, we can concerned. Uh huh. Why? Because they were busy here on on. And this when we can it could you 24 x and then minus huh? Trail Why? Over the tree where I&#x27;m under six axel Honest three z talent too.

Problem

Evaluate the indicated partial derivatives at the…

Problem 28 Easy Difficulty

Evaluate the indicated partial derivatives at the given points.
$$z=x^{2} \cos 4 y,\left.\frac{\partial z}{\partial y}\right|_{\left(2, \frac{\pi}{2}, 4\right)}$$

Answer

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