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Find (a) $f_{x x}(x, y),$ (b) $f_{y y}(x, y),$ (c) $f_{x y}(x, y),$ and $f_{y x}(x, y)$ for the function defined in the given Exercises.Exercise 4

(a) $\frac{-4 y^{2}\left(3 x^{2}-y^{2}\right)}{\left(x^{2}+y^{2}\right)^{3}}$(b) $\frac{4 x^{2}\left(3 y^{2}-x^{2}\right)}{\left(x^{2}+y^{2}\right)^{3}}$(c) $\frac{-8 x y\left(y^{2}-x^{2}\right)}{\left(x^{2}+y^{2}\right)^{3}}$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Idaho State University

Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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Find (a) $f_{x x}(x, y),$ …

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For the following exercise…

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In Exercises $1-4,$ find t…

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Find $f_{x x}(x, y), f_{x …

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so if you haven't done exercise for at least watch the video on it, I would go in, watch those first, Um, because we go through how to get these two partials first. Um, and we're just going to use this kind of bootstrap us into the problem. So first, if we want to find the second full partial work, respect tax will just take the partial over here with respect, Tax. As I remember, we're going to assume that why is a constant. So when I write this over here, I can pull the four y squared out. So before why Squared Del by Dell X of X over X squared plus y squared squared and we go ahead and take the derivative of this Now which we're gonna have to use quotient rule. So it be four. Why squared and then So it's low d hi minus high de lo and then all over the square of what's below. So it would be to the fourth power. Now on. I always have to say that little demonic. Otherwise, I just kind of put things all over the place. Um, so we take the derivative respect X, which is one and then over here, taking the derivative of this we need to use chain and power rules would be to x square. Plus, why squared that? We take the derivative on the inside. But remember this Why we assume is a constant So we just take the derivative of X word, which would be two X Um, now, Now we actually factor out when these expert plus wise words So that goes away, this goes away and then this down here becomes a three Onda. Let's write out what we have. First of this is going to be X squared plus y squared minus that we multiply all of that together that should be or X squared all over x squared plus y squared. Cute. Then I'll simplify that. And that should give us a y squared minus three x squared all over X squared plus y squared. Cute. Alright on then. I just kind of like this is one fraction just to make it look a little bit prettier. So I don't have those big brackets or mixed or are mixed, but our second full partial with respect X, it's going to be four y times. Why spurt minus three Expert all over X squared plus y squared. Cute. Now, when we take this next partial, um, you'll see how it's going to the kind of look very similar to what we just did. So there's gonna be this kind of symmetry, and we could go ahead and just take the derivative just like we did before. But I'll show you a little trick that you can do, um, to kind of save some time. Um, so we have the second full partial here. Now I can pull out the negative four x squared because remember, something X is a constant would be negative for X squared del Beidle. Why of why over x squared plus y squared squared. Now notice this and this the only real differences in the numerator. Here we have an X, and over here we have a why. So if we were to come down to this point because this is pretty much when we finish the derivative and just replace all the exes for wise, kind of just like switch them all that will give us what are derivative is, um so let's go ahead and write. That also would be first negative for X Square and then times. So instead of why squared is going to be X word instead of minus three, Expert is going to be three y squared and then down here instead of export to B y squared instead of y squared is X squared. Cute. And so, if you were to actually go through taking the derivative of this, you should even dope with exactly this here. Um, yeah. So any time you can see this kind of symmetry in some sense, like you could just like we did hear, switch all the variables and it will save you a bit of time from actually having to calculate everything. Um, yeah, on then, the only thing I'm gonna do is distribute this negative here. And if we do that, we would end up with it would be four x squared time. So three y squared minus expert all over. And then I'm also going to switch the x and Y down. There's expert plus y squared. Cute. And then this is going to be our second full partial, uh, with respect to why. Okay, Now we want to take the mix or find the mixed partials let me actually scoop these down a little bit. And so I would say the majority of the time you could use either of these two get the same answer, but we'll just go ahead and do both to just kind of make sure that we are getting the same thing. So over here. Um So let's get rid of that. Get rid of that. And now I want to do the make special respect. Why here? So this will be FX of why eso now remember, the X is going to be constant. So this would be four x del Beidle. Why of why squared over X squared plus y squared squared? And again we're gonna have to use question rule s 04 X um so low d hi Minus high De lo But I wrote the same thing twice. Even saying the demonic I still messed up. Unfortunately, um, so low d hi Minus high de lo and then all over the square of what's below. So in the denominator, we should still have the same thing. Eso Now why squared is just going to be, too why and then this again we would end up with so again, resuming X is a constant s 02 x squared plus why squared then derivative of X squared would be zero y squared would be to why, Okay, eso again, We can factor out this expert plus y squared like that of three. And then I can also factor out to why from here. So let's go ahead and do that. So it would be, too why and then times x squared plus y squared, um and then minus what do we have here? So to just be what or two Why squared and then all over x squared, plus y squared Cute. So let's go ahead and simplify this town And again I'm gonna hold off multiplying these because we're going to end up with some symmetry, just like we did before. Um, and if we can be a little bit lazy, uh, strongly a good idea to do that. So too why? And then that would give X squared minus Weiss word. And then this would be all over expert. Plus, why squared cubed? All right. And then at this point, I'll go ahead and multiply all that together. So it would be eight x y X squared minus y squared all over X squared plus y squared Cute. And then So this will be our first mixed partial and assuming we do the derivative right over here. So let's go ahead and write that down. So we would do Del by Dell not why, but X so that would give us f y x so we can factor out that negative for wise, we have negative for why here and then Del by Dell X of X squared over X squared plus y squared, squared and again notice that this here is essentially the same thing that we have over here for our derivative but the X in the numerator, Onda y over here. So again we could just switch all of the variables that we had down here just kind of make it a little bit easier on our lives. So, you know, I don't know. Let's just say we worked really hard to get all this and then this is where we kind of would stop so this would be equal to so now on the outside. So on the outside, we should still have that negative for why so negative or why But now this is going to be multiplied, but we're going to switch all of the variables. Um, so instead of having why, there that should be an x 02 X And then why squared minus X squared all over y squared, plus x squared, cubed And then we just multiply everything together and I'll distribute that negative on If we do that, we should end up with eight x y x squared minus y squared all over. And then I'll flip those again X squared plus y squared Cute. And so then you can see are mixed Partial here ends up being the same as well. So yeah, I would say most time you really don't need to do both because it should, for the most part, always give you the same answer, at least for most things you work with. Um, yeah, so it never really hurts to double check. So

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