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Find the partial derivatives with respect to (a) $x$ and (b) $y$.$$f(x, y)=\frac{x^{2}-y^{2}}{x^{2}+y^{2}}$$

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

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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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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05:03

Find the partial derivativ…

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01:49

Find the first partial der…

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05:27

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03:33

02:44

00:59

Find all the second partia…

So remember, in order for us to take the partial derivative of something, um, we assume all of the other variables are going to be constant. So over here, when we do del by Dell X or the partial respect X, we assume that these two wise here are going to be Constance. So, um, on the left, we would normally right, this is f sub extra, the partial derivative of that foot perspective ax. And over here, well, we have a function with respect to X and new Marie and denominator so we can use questionable. So I always have to kind of break this out off on the side because I always forget otherwise. DVD X Oh, hi Over love. So this is low d high. What makes friends Kriegel Ideal Low or load The I minus high Do you love all over the square of what's below? Yeah, so we will have just x squared plus y squared times. I'll just write this out. Dell by Dell X of X squared minus y squared at a minus, X squared minus by squared time still by Dell X of Expert plus y squared. And then this will be all over, but we have denominator squirts of expert plus y squared squared. Now remember, why is a constant with prospective X? So when I do, why square that's still a constant. So the derivative of this should be zero, and then we just take the Bureau of X squared like we normally So let's just be two x minus zero And then over here well, same idea applies X where we just use power. Rule two X And then why square That will just be zero when we take the derivative okay on DNA. Now we can go ahead and just tribute to access. Or better yet, we just factor the two x out since both these air being multiplied by two X So this would be to x times x squared plus y squared. And then we distribute that negatives would be minus X squared plus y squared all over x squared plus y squared squared. Now these X squares cancel out with each other, and we have y surplus Weiss Word or two y. So this looks like it ends up becoming or x y squared all over x squared, plus y squared squirt. So then this is going to be our partial to revenue with respect to you, X. Now, in order for us to take the partial derivative with respect to why let me go ahead and rewrite this so f of X Y is equal to x squared minus westward over x squared plus y squared. So when we do del by Dell, why now We treat these axes as if they're Constance. So over here on the left will be f sub y is equal to, um So again we use the question will also be X squared plus y squared del del by Dell Why of X squared minus y squared and then minus lo de hi minus high de lo So Dell, by the why of X squared plus y squared and then all over the square of what's alone. So again, remember this X and this X here. When we're taking the derivatives, we assume they are constants, so there's gonna be zero minus two. Why? And then over here, this will be zero plus to lie. Um, in this case, we could factor out a negative and two, Why from these? So let's go ahead and do that. So this would be negative to why X squared plus y squared and then plus X square minus twice word all over Expert plus Spice Bird squared. Oh, and maybe I should say where I got those negatives from. So we have negative two y here. We have a too wide and we have, like, a negative right here. So that's where I was pulling those out from, uh, and now the wise words cancel out. And then we have expert plus experts of two expert so we can rewrite. This is negative for Y X squared all over expert plus wise words. And then this here is our partial with respect to why and so again, just remember, whenever we're taking these partials, we just assume all the other derivatives or all the other variables are going to be treated as a constant other than whichever Wilmer trying to find the partial with respect to you

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