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WE
University of Notre Dame

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Problem 24

Use the Chain Rule to find the indicated partial derivatives.
$P = \sqrt{u^2 + v^2 + w^2}$, $u = xe^y$, $v = ye^x$, $w = e^{xy}$;
$\dfrac{\partial P}{\partial x}$, $\dfrac{\partial P}{\partial y}$ when $x = 0$, $y = 2$

$\frac{6}{\sqrt{5}}, \frac{2}{\sqrt{5}}$

## Discussion

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## Video Transcript

weather on this problem. We have a function p of a P equals square to use weapons V squared of w squared. Let's start right away by rewriting that as 1/2 power. Since we'll be taking derivatives, we also have you equals X times e to the y and B equals why you're the ex and w goes either the ex wife. Okay, let's make a little tree diagram for ourselves in the corner, just like the book does. So we have function p, which depends on U b and W and each of those depends on the two variables X and y Thanks my x Y and X y separates visualize things as we take derivatives. Okay, um, we also know that x zero and why's too okay? And that should be all we need. We will, as long as we're at it. Let's I get you ve N w. So, um, at a certain point, when x zero unwise to weaken quickly, figure out you is going to be zero uh B two times 00 is too, and ah e to the x y edo zero, which is one. So we have our values of U V and W at this certain point when we're ready to plug in for those. Okay, so let's begin. The first thing we're asked to find is partial of P with respect to X. So it's following our tree diagram we're going to get First of all, he goes to you and you w and w those two x that we haven't missed anything. Okay, we plug in as best we can, the partial of p with respect to you and so do this one first. Well, just looking over here at our definition of p, we're gonna have 1/2 times everything in there that the negative 1/2 times to you That's our partial of P with respect to you and impartial of you with respect to X Oh, well, let's look at our definition of you here so you can see the partial of that is going to be, uh, just eat of the why. And so we're done with our first term for their middle term partial appeals. Respect to you. Uh, the rather is going to look just like our personal feuds. Respect to you. Except it's the V, Mr Revenue. If we take on the outside. So we end up times to V instead of to you and the partial of V with respect to X, Let's look well, I would just be a constant so end up really with just itself again And finally Same thing. Well, partial p is respected w 1/2 your squared for speed square w squared to the minus 1/2 times to W and personal of W With respect to X now I'm gonna fly is a constant we're going to get Why eat of the X y from the King girl. Okay, so at this point is just a matter of plugging in values that we know and we do know all the values. So let's just be careful as we plug in May of 1/2 times. Let's say you a zero V is two squares for W is one so foreign one beginning of five in that parentheses. Okay, good times, too. But you is zero. So actually, he can't stop. Is this entire first term whatever we're going to get for you to the why did This whole first term is gonna disappear because of this zero here. Good. So the second term 1/2. Let's look at five in the parentheses. Times to V. V is too. Why is too e to the zero. Go ahead, write that even know that will become one very soon. And finally it is Last term again. 1/2 five to the negative one half, uh, times to w equals one. Y equals two p to the X Y zero. So we'll get that. Okay, so you're almost there. But we have 1/2. I will have a lot of canceling here. So the 1/2 1 of those two goes away. Get so then five to the negative. 1/2 is one over the square. Five, we'll have two times two looks like four on top and our last term here once again, something over the square root of five. We, everyone half times to that, cancels and all that's left is that too. So final answer for this one six over the score to five. That's partial of P with respect to X. Okay, so now partial p with respect, toe, why will be much the same. So we'll erase some things, but not too much here because, all right, everything in blue is going to stay the same. Because if we're going down the tree, we still have partial of p respect to you and V and w so will erase all the arithmetic in our last two steps, but not too much else. So let's think about what we're gonna change. All that's gonna change is that instead of with respect, X, we're going to be with respect to why, Okay, so that means, uh that all these exes here will become wise so partials. But why? Why and why? So then the blue parts do not change all that change. I'll raise Thies green parts. Those are all that's going to change. Okay, So partial of you with respect to why let's look so x is like a constant. So we will get X. You know why? Good? Um, partial of V. With respect, toe. Why this time X is the constant. They're good. And then finally partial of W with respect toe. Why? Well, if X is a constant than by the chain, really get X e to the X y. Okay. And once again, we plug in. But we're used to this 1/2 the middle of the parentheses is five, as it was before time, too. U is zero like it was before. So again, whatever that is, this whole thing is gonna be zero. We don't have to worry about it. Right? Plus 1/2 5 again. C two v was to you. The X is eat of the zero. That'll be one. Finally, 1/2 five. None of this. Any different times to W equals one X equals up zero. So, uh, this whole term will also disappear. All that's left of their middle term. So we have that squirt of five on bottom. Just 1/2 of the to cancel out. We're left with just a two time to one on top. So our final answer There's two over the square to five and we are done. Hopefully, this helps.