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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 7

(a) $240 y^{3}\left(2 x^{2}+3 y^{4}\right)^{3}\left(6 x^{2}+y^{4}\right)$(b) $24 y\left(2 x^{2}+3 y^{4}\right)^{3}\left(4 x^{4}+192 x^{2} y^{4}+759 y^{8}\right)$(c) $240 x y^{2}\left(2 x^{2}+3 y^{4}\right)^{3}\left(2 x^{2}+19 y^{4}\right)$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

Oregon State University

Harvey Mudd College

Baylor University

Boston College

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 $3-20$ , find…

So if you haven't done exercise seven year or at least watched video, I would go do that first because we're just going to use the work we did there to kind of bootstrap ourselves so we don't have to find these partials again. Uh, so first, let's just go out and find the second full partial for prospective X. So in order for us, it is from a reason this Hawaii here is constant, so we could factor out that white cubes would be 80. Why Cube del by Dell x of X Times two X squared plus three y to the fourth race the fourth. And now we have two functions that depend on X being multiplied together. So we use product rule. Should be 80. Why cube times on, I'll just kind of write this out. So two X squared plus three wide to the fourth raise the fourth del by Dell X of X plus Ex del by Dell X of the derivative that right? Actually, I probably need to skip this down quite a bit. All right, now, the derivative of extra just be one derivative of this would be so we use power and then chain rules would be four times two X squared plus three y to the fourth Cube. And then we take the derivative on the inside. Remember? Assuming why is a constant so that would just be zero. And then two x word would be or X. All right on that. At this point, since we have three of these, at least in each, we can pull those out. So to be 80 y cubed and then two X square plus three y to the fourth Cute. And then everything else on the inside. So we have two x square plus three y to the fourth. Um, actually, let me just cross that out that way. I don't accidentally put it together, so it would be four times four times x squared. So that would be plus 16 x squared. Um, and then I would give us 18 x where it looks like, so we would have any Why cute two X squared plus three y to the fourth cube and then 18 X squared plus three y to the fourth. Um, I guess we could also factor how the three from this, I just go ahead and do that as well. So factor that three out from over here. So to 40 y cute two x squared plus three wide fourth cute. And then X word for about X six six x squared plus y to the fourth. And so now this is going to be our second full partial of respect tax. Um, And now we can go ahead and find the full partial with respect to why? So let me think. Uh, and in this case knows how we actually have three functions, all depending on why being multiplied together. Um, so what I'm going to do first is actually distribute this 12 y squared over here. So two x squared plus three y to the fourth race to the fourth. And then or very I'll keep the 12 out front. Um, they will distribute the white squares to be two x squared y squared plus 23 y to the sixth. Okay, And now when we come over here to take the partial of this with respect to why would end up with so f why 12 on the outside, And then again, we use product rule, so we'd have to x squared plus three y to the fourth, raised to the forced del by del y of two x squared. Why squared? Plus 23 wide six and then plus, Then we just reverse the order again. And then Del Beidle, why of two x squared plus three wide to the fourth race the fourth. All right. And then we just take the derivatives of these assuming X is a constant. So this excess a constant here. So that would be four x squared. Why? And then that would be eso 23 times six is 1 38. So that would be plus 1 38 y to the fifth hour. Um, and then over here, if we take the derivative of this, um, it will be essentially what we did before, but now we assume X is constant. So power and changeable smoke, this first part will look the same. But then when we take the derivative here, that would just be zero. Then we'd have 12 y cute. There. Try it. Um, And again, we can factor out three of these. So let's go ahead and do that first. So this would be 12. And then times two x squared plus three y 2/4. Cute. And that would then give us just two x squared plus three y to the fourth times or X squared. Why? Plus 1 38 wide to the fifth. Andi over here. That would give us 48 y cube times two x squared y squared plus 23 y to the sixth. Um, then is there anything that we can kind of factor out easily from thes to maybe make this a little bit easier toe work with, um, I think we could possibly pull out a maybe, just a Why? From everything. I really don't see anything else I can really pull out. Um, so I can, like, factor out. Well, why from both of these and then one of the wise here, Um, and doing that. So actually just moved that out front here, actually. So that why goes way this becomes to the fourth power, this becomes squared. And then, if I were to pull out a Y cube from here, I really don't see that doing much to simplify it. Yes. So the only other thing I can really think of doing is like distributing everything and then, uh, make it a little bit. Messier. Um so let's just actually go ahead and foil this a lot. Or actually, I guess we can also pull out of two from everything. Eso I can rewrite this out here as 24 and then this becomes to, um and then 1 38 divided by 2. 69. And then over here, this would become 24. Oh, yeah. And now I really don't see anything else I could really do with this point. Unfortunately. So yeah, let's just go ahead and expand everything out that supply it down s o B 24. Why two x squared plus three y to the fourth. Cute. Um, actually bury it. Let let's just leave it like this. Um, I mean, at this point, it's really just doing some algebra to kind of rewrite and simplify things down. Um, and we're here more so for the algebra than anything else. I mean the calculus than really the algebra. So I'll just leave it like this even. I mean, I guess we could make a little bit prettier, but it would just take a lot more time to do so. So I'll just leave it like that for now. Yeah, Um, now they want to find the partials of these eso. When I'm looking at both of these, um, it doesn't remember which one we do. We should get the same one regardless. Eso for some of the other ones. I did both, but since this one is actually kind of fairly complicated for both of them, um, I'm only going to do one. And the one I'm going toe do the partial off. If we're comparing like this here, the partial would respect toe. Why versus appear the partial with respect to X. The partial with respect to X, to me looks a lot simpler. Um, so I'm going to take this one and then try to take the partial put respect toe. Why now and then that should give us the same thing. So let me get rid of this. Get rid of that on the outside. So now we'll do del by del y. So I give us f of x y, which again should be the same thing as f of X y. It doesn't really matter which one. We do a t East in both cases. So, uh, that means it's X here is going to be a constant, so we can go ahead and factor that out. So it be 80 x and then del by del y of why cute? And then we have two x squared plus three y to the fourth race to the fourth right. And now we just use product will just like we did before. So a t X um, look two x squared plus three y to the fourth race, the fourth del Beidle y of White Cube. And then just put those of why cubed del by del y of two x squared plus three y to the fourth race fourth. Now why Cubed would just be three y squared and then the derivative of this over here, um, we would use chain and power also four to expert plus three y to the fourth Cube. Take the derivative of the inside. But remember, we assume this excess constant, so just be 12 y cute. Okay, um and now so again, we could factor out three of those copies of this. So let's do that to be 80 x and then two x squared plus three y to the fourth. Cute and we could also factor out three y squared to be three y squares that goes away. And then this over here would just become four. Why, Um, yeah, that we could multiply everything together. Eso over here. This should just be we could that blue So two x squared plus three wide to the fourth and then plus so four times four times y so that would be 16 y to the fourth. And then we could just go ahead and combine those white to the force on actually, everything out here as well. So would be to 40 x y squared two x squared plus three y to the fourth cube and then two x squared plus 19 y to the fourth. And so then these are going to be are mixed partial or this is going to be our mixed partial. So again you could also do the other one. But it just looks a lot more complicated to do aan den just kind of for sake of time for the video. All kind of leave it off here. But again, remember, in most cases it doesn't really matter which way you take this partial, you should get the same thing. Progress

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