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For $f(x, y, z)=4 x^{3} y^{2} z^{2}+4 x^{2}+2 y^{3}+5 z^{5}+3 x-2 y+11 z+12$ find $f_{x z y}(x, y, z)$ and $f_{x y z}(x, y, z)$.

Both are $48 x y^{2} z$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

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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So they want us to find the partials. So of f of x, y z and then of f x Z y. But we actually have a the're, um, in this chapter that says it doesn't really matter what order we do these in, especially if these air all going to be kind of continuous in some sense or smooth functions. And so this is essentially kind of like a three variable polynomial. In some sense, this will always be smooth, so these two will be equal to each other. So if we find one of them, then we've actually found the other s. So I'm just going to do it in this first case here. Um, so let's just go ahead and do that. So the Dell by Dell X to start. And so remember, we're going to assume all of the variables other than X are constant. So when we're looking through this, um, this y cube times too well, the derivative that would be zero, because we're assuming it's a constant same thing with Z to the fifth times five, um, two times why 11 times see, and then 12 is also just going to be zero okay on, then we can just factor out this. Why squares eastward? Just like we with the four when we take the partial. That and then just take the derivatives. Everything else. So let's just go ahead and write that out. So f sub x is going to be so actually, just write this out for so four. Why Square c squared and then Del by Dell X of execute and then the derivative off four x squared, we would use power. Also eight x derivative of three X would just be three. And then, um, taking the derivative, that would be three x squared. So you can go ahead and multiply all of that together now, so that gives us our first partial, but respect X would be 12 x squared y squared cease word. Okay. And now, so again, it doesn't matter if we take the partial with respect. Why? Or with respect Dizzy from this point s. So I'll just do it with respect to why So that would give us f x y is equal to So remember this X and the sea we can pull out. So we try this 12 x words eastward. Del by del y of y squared. And then this ends up being too. Why so multiplying those together that gives us 24 x squared y z squared. And then lastly, we could go ahead and do the partial of this with respect to Z Ah, and so that would give us four x y z burn up for X f x y z for those partials. Um, And then again, just pull out the expert. And why? Because we're assuming those are constants with respect to Z at X Z, and then that would just give us to Z, Then multiply all that together. And so we get this here would be 48 x squared y Z. This is going to be our solution. Um, and so again, if you're well unsure about why we could do this, we could also just go ahead and do it with the other one. But this should also be equal to f x z y. But let's just go ahead and maybe just try the other way just to kind of confirm that is the case. So if I come over here and instead do del by del Z, first of this so again that would give us so perhaps of ecstasy. And then we can pull out the X squared and the white squares would be 12 x squared y squared del by del Z of Z squared, which this is going to give us to see multiplying everything together would give us 24 x squared y squared Z and then lastly, do del Beidle. Why of this partial and then f x z Why is going to be so again we could pull out the X in the Z So Dell Beidle high of y squared and then that's just going to be, too. Why multiply everything together? And so you see, we get 48 x squared y z, which is exactly what we have got before, say again. As long as it seems like it's going to be a smooth function. You don't need to check both of these, uh, always for practice. It never hurts to actually show that they do equal each other

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