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University of Notre Dame

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Problem 16

Suppose $f$ is a differentiable function of $x$ and $y$, and $g(r, s) = f(2r - s, s^2 - 4r)$. Use the table of values in Exercise 15 to calculate $g_r(1, 2)$ and $g_s(1, 2)$.

$g_{u}(1,2)=-24$ and $g_{x}(1,2)=28$

## Discussion

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## Video Transcript

Heather. So in this problem, we have a function G and the variables. There is a function of R and s, but we're also to think of this as a function named F. And it's two variables are ex, which depends on Arness and why? Which also does. So let's make a little tree diagram just as in the book. So our function f depends on X and Y, but each of X and y depend on our end us that will help us keep things straight. Okay, Also relevant. We care about the point when r and s is one common to both. Questions have to do with finding parcels of G When r and s are one and two, we're going to need to figure out what X and y are as well when are in Esther wanted to. So let's write down what these functions of X and y actually are so ex. Now, even this copy right out of the books Definition X in this case is too are minus s. And why in this case is s squared minus for our okay and no, the reason we want that will use it later. But now we can figure out that when r and s are one and two, you can figure out what X and y are. Just plug things in sonar NS and warning to let's see, for ex of two times one minus two. So two minus two is zero. That means X zero. And likewise for why? Let's plug in one for our two for s to scored us four minus four times 14 minus 40 as well. So now we know that at this point that we're dealing with are Nasser wanted to, But also, X and y are both zero, so that will be used. Okay, um, let's get started. And if we need partials along the way, we can take them as we need them. So we are asked, First of all, to find g partial are at the 0.1 shoe. Okay. And now we can use our tree diagram, so we want to get down to the variable are so we'll have to go down both branches. So starting on the left, that f x means of partial X and, um and then the point we're taking this at because F is a function of X and Y our input, sir. 00 there. And then you want x six of our X partial are, um and now we're into R and s coordinates because that's what X is given. And so were and wanted to likewise the other side of the, uh of the tree diagram It goes toe. Why? So that means of partial Why, at that 0.0 times, then why partial are at the 0.12? Okay, so let's see what we can do. F partial X at the 0.0 Well, you can just take that off of the table. So we look at the table they give us that comes out to four. So this just from the table is the number four for this next part X partial are Well, here is our definition of X so X, the derivative partial derivative with respect to our you can see just at a glance is just going to be too, um, independent of what r and R. R N s are. So in this case, it doesn't even matter. That airplane was one common too. All right. Plus, once again to the table for F partial y +00 in the table says that is the number eight. And then finally, for why partial are once again we can use our definition over here of why? And we can see again a glance the derivative with respect. Our is minus four. Uh, just another Constance. Okay, so we have our first answer four times two is eight, eight times negative. Four is negative. 32. So we should end up with minus 24 as our answer for that. The 1st 1 now to get G partial asked to do the exact same thing. So this should feel very familiar. You go down the tree again. So the only difference this time is we want to end at the variables s so we'll go down these green branches. So the end, it s so the n f partial x at 00 times X partial s I wanted to plus of partial y 00 times. Why? Partial s at 12 Okay, we can use table of values here, and in fact, these firsts Once these red ones, they won't change. We took those from the table before, so those are still the same as they were. This is four, and this was eight. The only thing that will change will be the partials of X and y, with respect to s Now instead of are so over here on the left, we're going to want Yeah, that's just must just keep track of those here also in blue. So as as X Partial s we look at this. Well, the derivative of two are minus s with respect to s is just minus one the constant and the derivative of partial derivative. Why, with respect to s wise s squared minus four are so we get to s and again we're doing this at the point one comma two. So it's negative for plus eight times to what is there s value. Well, at this point, it's too right at the point. One comment You rs variables right there. So two times two negative for plus, That's the eight times four is 32 which will give us 28. Hopefully, that out