find-the-partial-derivative-of-the-dependent-variable-or-function-with-respect-to-each-of-the-ind-56

Question

Find the partial derivative of the dependent variable or function with respect to each of the independent variables.
$$f(x, y)=\frac{	an ^{-1} 4 y}{2+x^{2}}$$

Tyler Moulton · Accepted Answer

Yeah, we want to find the first partial derivatives of the function F of X. Y is equal to the arc tangent of four Y divided by two plus X square. This question is testing our understanding of differentiation both from the techniques we learned in single variable calculus and for the purpose of partial derivatives and Multivariate calculus. The first part of it is we&#x27;re searching for our DF DX and DY to obtain each. We use standard differentiation taking from single variable calculus. Where for DF dX we differentiate with respect to X treating any white terms the constant and similarly for the FBI we differentiate with respect to Y treating any X of constant. So the only white terms are temper why? The only X terms are two plus X squared. So FX is arc tanne for Y times negative. 1/2 plus X squared times two X. This is using the chain rule on the denominator. Before we proceed to FY, Which I have written at the bottom here, we should note that because we use the chain rule denominator, this two plus X square needs to be square rooted or rather squared in the denominator via the chain rule or the power rule chain rule. Then, for Fy below, we note that the only term is our 10. For why? So are derivative is 1/1 over two plus X squared. This was a constant. Remember times 4/1 plus 16. 1 square. The directive are 10 for why? Or for over two plus X squared times one for 16. Why? It&#x27;s weird.

Linh Vu · Answer

thanks. Why you? Could you the one minus tension in verse off the X squared plus quite square Notified a partially riveters. But the X where do you think there were arguably the constant? So we have drifted on the wanted coaches zero and minus the riveting attention it could you want off the one plus x squared Plus quite square, totally square. And by the general manager terms they lived in the inside. So get in Quick unit, your ex near And now by symmetric we can in the Gender X and the Y Now, did you get the partial? Everywhere there was budget A Why? And we&#x27;re gonna mind us and will be true. Why now and then we have one plus x squared plus y square 70 square.

Sriram Soundarrajan · Answer

today we&#x27;re going to sort profitable tech victory here found share off X rays given by one minus tanning with off Take the square by square We heard two different She parted literally do off with respect breaks as the last right so fast Really friendship with respect to X It will be getting us Cirone minus one by going pressed excess card which will be hated square past life where the whole square and do with it comes to the n minus Do it one plus a three squared plus life Glad it up close similarly you know, departed out of it off the function f of X like we just picked away you Look at it is like Is it old minus blocking by one plus interest square. So it was quite is X squared plus place where the whole square indeed roulette. So it will be like minus two Lie devoured by one place. Take this class less lightsquared that whole square. Thank you

Sriram Soundarrajan · Answer

Hello. I&#x27;m going because our problem number 22 you have given f f enslave is you in by burning with interests whereby vice, Where so? Well, I wouldn&#x27;t be doing like we had to depression in respect of this to get the neckline bite Bless X squared. So every square by life square the horse square which is getting late Oh, squared and again to in Unless there is a place where so in begetting leg why has to for her by where is 24 plus x fester for Pinder who left diverted by life? But is it Gordo two X life squared, divided by excess to for press lighters to foot. Okay, then we heard a different you. Thank you with respect away. So what would be getting it like seeing like anybody plus a two squared bye bye square that over square can do it just kind of minus two Bye bye Q It should be getting like why arrested for by where is the fourth best existed for Indo minus two discredited by like you which is a goto minus two excusable life They were by why arrested for us existed for Thank you

Will Erickson · Answer

another in this problem, we were asked to find the second order partial derivatives of this function. So let&#x27;s start with the first order Partial derivatives. The start with s sub X will treat excess the variable and why? As a constant, we have inverse tangent of this inter function here. So we&#x27;ll follow the chain room Go from the outside in the derivative of inverse tangent we remember is 1/1 plus are inter function is why over X and that&#x27;ll be squared and I wear the chain rule we multiplied by the derivative of this inter function and specifically, in this case, the partial derivative with respect to X of that inter function. Okay, now, with respect to X this derivative since really this is the same as White Times X to the minus one. Hopefully that we&#x27;re we&#x27;re convinced will end up with minus y over x squared were just using the power rule in treating why the constant? So make sure you&#x27;re convinced of that fact. So we end up, let&#x27;s see, so we&#x27;ll have a minus y on top. You can put this X squared on bottom. So for multiplying export times one gives us x squared in X squared times this while I scored over Exc Word will just give us why squared So just make sure you follow the algebra on how that x squared on bottom canceled out this x squared on bottom Okay? And s So why now? Once again, we start from the outside in so it&#x27;s still an inverse tangent on over one plus why over X all squared This time, though, we multiply by the partial derivative with respect to why both that inter function Why of Rex with respect to wise? A little easier if why is the very blond ex is a constant miss derivative is just one over x so end up We just put the Exxon bottom So one times x x Why squared over X squared times x We&#x27;ll end up with y squared over x So again, Just make sure you see how this X here multiplied by everything on the bottom Here gave us X plus y squared over x not X squared of it. So we have our first order partial derivatives. Now, just time to get the seconds. Let&#x27;s begin with s sub x x So we look at some X, and we will take the partial derivative with respect to X again. So for this one, um, we could use the quotient rule. And why don&#x27;t we go ahead? You can also do product rule. Let&#x27;s do the quotient rule just to get to practice. So kosher rule tells us for training exit the variable will get the denominator times the derivative of the numerator. That&#x27;s that&#x27;s going to be zero, since we&#x27;re talking about the derivative with respect to X. And so there&#x27;s not an action, the numerator, and that&#x27;ll be zero all over the numerator times the derivative of the denominator with respect to acts which we can see his two ex all over the denominator squared. So on top of see, we have, uh, x squared close. Why? I&#x27;m not sure if there&#x27;s a to there, but there should have been So you&#x27;ve X squared plus y squared times here. Although so that whole first term disappears. All we have left on top that&#x27;s C minus minus gives a positive. We have two x y and then over this same denominator. Okay, so that is s a x X. Now let&#x27;s do s some Why? Why we Look, it s sublime down here and we take its partial derivative with respect to why, uh let&#x27;s see. So once again, let&#x27;s do the product rule here. Just are actually Linda being a chain rule. So in other words, but to rewrite s sub y as X plus y squared over acts all of the minus one power and I will use the general on this. So the minus one comes out in front. Keep all that inter function that seems it was power comes down to negative too. Now we multiplied by the derivative of this inter function here with respect. Why get the multiplied by? This thing is driven. If they respect a why But this is quick. This X will give us zero. And with respect to why the derivative of y scored over X is too. Why over exits exits just a constant okay. And we can rewrite that a little nicer as let&#x27;s see, So we&#x27;ll get negative to I all over x times X plus y squared over X squared. I&#x27;m sure some of the things you could do algebraic Lee with this, but this is certainly equivalent to the correct answer. Okay, time now for the mixed partials. So ex of X Y let&#x27;s look at some X here and take its partial with respect to why so for this one, that&#x27;s definitely use a quotient rule again. So we have the denominator x squared, plus y squared times the derivative, the numerator with respect to why so that&#x27;s gonna be minus one minus the numerator times, the derivative of the bottom with respect to why which is to I all over the denominators squared. Okay, so let&#x27;s see what we got here. We have minus X squared minus y squared and then we have a plus to y squared all over that bottom and we see minus y squared plus two y scored is actually plus y squared so we could just rewrite the top a little bit as negative x squared plus y squared. And there we go for that. All right, Come. Last thing. Did you know he wants the other mixed partial sa y X? There may be a question. Can we use, uh, Claire Rose theory on that serum to from this section? And in order to do that, we would need all of our original function and all these derivatives to be continuous. Um, And if you&#x27;re looking, singing, why over X when X is approaching zero, that&#x27;s not continuous, which is true. So to play it safe, you&#x27;re running. We actually work out as sub. Why? X So here&#x27;s s sub y found here and let&#x27;s take its derivative with respect to X. Once again, let&#x27;s rewrite this as X plus y squared over. Excellent. The minus one is not going to see Is the chain rule minus comes out in front If X plus y squared over X all of the minus two times. Now we have to take the derivative of this inter function well with respect to X so the ex becomes a one and this next part by scored over X. If we think of the power rule, it&#x27;s the same as why score times actually the minus one. So we&#x27;ll get minus. Why squared over X squared In other words, minus why scored acts to the minus to just make sure you do the power will on that if you&#x27;re not convinced and make sure you end up with this and just so we have room here that&#x27;s break out into this left half. So to clean this up in a way, we can see you get a minus on the bottom. You know what we&#x27;ll get? We&#x27;ll get that X plus y sward over X all of the second. Because of this minus two and then on top, we have a one minus my squared over X squared. And from there again, if you want to multiply through by X squared top and bottom to clear out the denominators, I certainly can, uh, see, um but for now, why don&#x27;t we leave it like this just so we don&#x27;t get too lost in details. And then again, if you want to do some algebra on this just to simplify to make it look like, uh, make a look a little simpler, you can. But the important thing is thes are the sex

Tanishq Gupta · Answer

in the question we have to find before second. Actually, they were before any worse off. It&#x27;s by our now moving towards a solution. Find before departure that before step to compute the first far sure that ever come off the function. So divorced, Given some benefits, right? Was 10. Invoice. That&#x27;s our guy we have. Then f way they&#x27;re less was to one boy, one less instant, like with all the sweat in the square advice. But I will be being exists. Vice. Where, upon one plus x parts expired with powerful singularly de l f I held by were required to extend part three by on less experience. Six. Why being a computer before second buyers showing any weakness? So we&#x27;ll be in for that Dellis were f l exist. Where will be for you? Six X vice Where less into one place extra pass six by two Powerful minus three X squared by square into six x two par five Quite with our four upon one less expensive 56 into life with the bottle for this, but this will be 4 to 6 x wise where minus 12 x to the power 72 by two the part. Six Born by blessed, extended by six by for the car phone poles that will finally for six x Why sweat one minus extreme power. Five. Try to divide full Poland one less x 25 6 White 54 This is a first second are sure that we do no moving for that. There is a f on l&#x27;ve, I swear will require extra 31 less expert six by to depart for minus two x Q by four x six by partly upon, well, less extended by six quite by six whole square. No, this will find a leak or who it&#x27;s you won my nest two extra by 6.4 upon, one less. It buys things flight to depart for holds. This is a second part of this. Second are sure that we were going away for the X they&#x27;re like upon into do. It&#x27;s to buy upon 16 by six and right to the Final Four. That would be six x squared by one less X require six. By to depart for my next two X, you buy into six. It&#x27;s by five right to buy food upon, one less experts exploited for elsewhere. This would be finding work through six Texas. Where why, when my nest through its it&#x27;s expired two by four born one. Yes, extra buyers expired to the body full of booze O in fort that they f four tell by their next it&#x27;s well, we were 10 Fornell by three Exit Square Why X Squared Born One less extra 65 Part four which will Week six Xs go and buy one last extra sex life, in part for minus three. Exists when? Why, where whole excellent. Six by three upon, one less extra by six life in part for whose? Which would be finally word six. Sexist by one minus. Who extinct by sex upon one less experts excited for old sweat and SPF the full, different second Parcher that so this is you find a lot

Problem

Find the partial derivative of the dependent vari…

Problem 12 Easy Difficulty

Find the partial derivative of the dependent variable or function with respect to each of the independent variables.
$$f(x, y)=\frac{\tan ^{-1} 4 y}{2+x^{2}}$$

Answer

$f_{y}(x, y)=\frac{4}{\left(2+x^{2}\right)\left(1+16 y^{2}\right)}$

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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$f_{y}(x, y)=\frac{4}{\left(2+x^{2}\right)\left(1+16 y^{2}\right)}$

Basic Technical Mathematics with Calculus

Lily A.

Anna Marie V.

Caleb E.

Joseph L.

Vectors Intro

Vector Basics - Example 1

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

Lily A.

Anna Marie V.

Caleb E.

Joseph L.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus