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Find (a) $f_{x}(x, y),$ (b) $f_{y}(x, y)$ at the indicated point.$$f(x, y)=2 x^{2} y^{3} \ln \left(x^{2}+y^{2}\right)(1,-2)$$

(a) $-32 / 5-32 \ln 5$(b) $64 / 5+24 \ln 5$

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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So this is kind of a continuation of number 15, because the number 15, we actually found the partial derivatives with respect to both X and why here? So if you've already watched those videos, I'm just going to go ahead and kind of skip over and just kind of write down the results that we got from those on it. For some reason, you haven't watched those. I would go back and watch those videos first, um, or at least try to do the partial and see if you get the same things that we got here, but just kind of save some time. Since we technically have already solved this, I'll go ahead and just apply this. So if we want to find partial derivative at a particular point, all we need to do is take that point. It kind of just plug it in just like we would for the single variable driven. So we do f sub x of one negative two. And then remember, this is our X value. This is our why value. So we just plug these in over here, so it be four times one of the negative two cubed, which would be negative. Eight. And then this would be all time. So x squared. So that would be 1/1 squared. Plus, then we swear to so there before, plus natural log of one plus or that would go ahead and simplify. That s o. That would be negative. 32 times. 1/5 plus natural log five. Um, I guess we could leave it like this. Bob is going to distribute the negative. 32 would be negative. 32/5, plus negative. 32 natural log of five on. So then this is going to be our partial derivative at the point. Negative or at the 0.1 negative, too. And then over here for a partial. Uh, why at that point, same thing. We're gonna plug in the point. So we have f why one negative too. So it would be to one square, which is just 12 squared four. Yeah, And then we have to and the negative two squared. So four. And then the expert plus y squared. Well, we actually found that over here is just five. So I just write five and then plus three natural log of five. Then we could just go and simplify that eso It would be eight times. And then eight. Fifth plus three natural log five. But then we go ahead. Tribute eight. So it be 64 50 plus 24 natural log of five. And this here is our personal at the 0.1 negative too. Uh, So again, if you haven't done problem number 15, I would go watch that so you can see how we actually got these two partial derivatives here.

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