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University of Notre Dame

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Problem 22

Use the Chain Rule to find the indicated partial derivatives.

$ T = \dfrac{v}{2u + v} $, $ u = pq\sqrt{r} $, $ v = p\sqrt{q}r $;

$ \dfrac{\partial T}{\partial p} $, $ \dfrac{\partial T}{\partial q} $, $ \dfrac{\partial T}{\partial r} $ when $ p = 2 $, $ q = 1 $, $ r = 4 $

Answer

$\frac{\partial T}{\partial r}=\frac{-1}{32}+\frac{1}{16}=\frac{1}{32}$

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## Discussion

## Video Transcript

either. So in this problem, we have function t just to find as cv over to you could be, in other words, make a true diagrams in the book. T depends on the variables you envy. Okay, But also, U and V depend on the variables. Uh, P Q and R P Q r. There's a tree diagram in the corner. It's also right what u and V are. So you equals Peak, you squirt of our let's re write that is part of the 1/2. Since this is calculus and the equals p squared of Q. You are okay, um, and finally, let's remember P is too. He was one r is four. So you somewhere here, this is make a note that p Q r are to one and four. Okay, so that'll be useful from So at this point, you should be ready to start. And the first thing we were asked to find is, uh, see partial tea with respect to pee. So why don't we use our tree diagram and we want to end that the peas down here so we'll go down the left branch first, which gives us All right, So we go from T to you and then from you to pee. Plus now the right side of the street. You go from t to be come in for a V to pee. Okay, so one thing at a time here, let's start with partial of tea with respect to you. And for that, let's just look at our definition of tea. So, um, partial tea with respect to you, let's use the quotient rule on tea over here. So you was air variable as we're considering it. Let's use the quotient rule. The caution rule says it's really good. Low. So to you, close v, the derivative of the numerator. Well, the derivative of the with respect to you is zero minus two. The numerator, which would be the derivative of the denominator, the derivative of to you with respect to you is just two. All right, and then write all over over the denominator squared. Okay, So they're d t to you for do you with sure. Still went there. Okay. Partial view of respect to pee again. For that, we can just look at our definition of you here. And if p is our car variable the other two are constants, So Ah, we will just get Q when that squirt of our that's our derivative of you with respect to pee came Plus same thing over here. Partial deal with respect to V. Once again, we can use quotient rule. So we get nominator. The derivative of the numerator now is one. Because we're going with respect to V minus. The derivative of the denominator is also one with respect to be over the denominator squared and finally, DVD p We can just live it R V over here and once again we'll get, uh, a similar thing is before you don't have times are that's the derivative of the With respect to P A partial later. Okay, so we have our answer. If we just know what to plug in, could notice that this is zero. And what are we going to need? What we're going to need? Um, we have Q p and our values from this point down here. We're also going to need you and V values at this point. So let's make a note. Are you at this point equals Will P is two. Q Is one in the square to four it will be too. So two times one times two is four and V will be, Let's see two times. Now we have one of the one have power is one times for we have two times forwards is eight. Okay, now we can plug everything in. So we get so you ve was 88 times two is 16. So we get negative 16 on top to you plus V two times four is eight plus eight is 16. So we get 16 square there times. Q Is one Are is four. Okay? The orders. Plugging things. And now on the other, other big fraction here to you plus V two times four is eight plus eight is 16 minus. He was eight. Our denominator will still be 16 squared times. Skirt of Q is one and R is for all right now we're getting somewhere. Let's see what we have. A negative 16 over 16 squared. Well, of course, that will give us a negative one over 16. It's hard for us to 16. Minus eight is eight times four over 16 squared. So let's see. Getting negative 1/8 from our first for action. But for a 2nd 1 It looks like we're also getting 1/8 when the dust settles here. And so we can, uh yep, that we can get her final answer here of zero. Sorry if there is a pause in. They're not sure it's video post or not. Um, so we have partial tea with respect to pee. Um, okay, so the other two will be just the same in spirit. So it's a race, Uh, what we had. But we'll start from same idea here, and we'll use all the information on the left, so this would be a little quicker. So, um, we now are supposed to get partial of tea with respect to Q. So when were placed this down here with a Q. And listen, if you go down the tree, we'll get harsher lived, Chief to you and you. You good on the other side of the tree, the yank. You okay? Good. Um, personal tea with respect to you, because we had that before. Uh, okay, so again, quotient. Rule. Um, what if we If we recall, we had denominator times zero minus numerator derivative of the denominator two over denominator squared. Now D u d Q. Let's look over here you again. So we're going to get P. Times are the 1/2 plus C for the next one with prospective E. Parshall of Tea will give us again by the quotient rule denominator driven of the numerator is one minus numerator. Derivative of the denominator is one is, well, all over. Denominator squared and DVD. Q. Um, for this one with a little more careful now with respect to Q DVD, Cue the P and the R. Constance. But then we had Q to the 1/2. So the 1/2 comes in front will get cute of the minus 1/2 so it has to be cautious. There 1/2 P. R. Q to the minus one. OK, on we plug everything in and see what we get. So, um, as before, when you plug in that zero there, we'll get negative 16 on top of our big fraction. Uh, and we'll get 16 squared on the bottoms. That has not changed. P is, too. Are is four score two, that is two Next one to you plus V uh is 16 minus the eight all over 16 squared times half He is to our four Q to the negative 1/2. He was one, so that will still be one. Okay, so then we have C 1/16 times, four B, negative for 16th plus 16 minus eight. See, what else do we have here? We've got, um, thes cancel out. C minus eight is eight eight over 16. Squared times four. We can think of this as eight times. Two times two. So once again, it looks like we're going to have negative four sixteen's plus 4 16 which again is zero. Okay, um, so our last the last partial we have to get is a partial tea with respect to our And to do that let's be a little smarter than what We're haste this time, because d t d u and D tv, those will not change all that. All that will change will be these partials. So it's a race that Oops. You might as well actually put some of those back. Um, yes. Of these fractions will stay the same. All that we need to change our these parcels on the afterwards. Here. Okay, Good. In other words, all that will change. Are these right there? So it's a race those, and we'll finish up the problem. So over here we're now supposed to get, uh, partial of tea with respect to our And that means this will become a partial view with respect to our partial V of respect our so partial of you with respect to our again. Look at the definition of you, but we should get P. Q. Now we need times. 1/2 are the negative 1/2 and the partial of V with respect to our is just P Q. The 1/2. Okay, so plug in as before. P Times Q. Two times one Just two uh, times half times are to the negative 1/2 ours four. So Ford of the negative 1/2 is one over the square root of four. So that's 1/2 then over on the right, P is to Q is one. That's all we're gonna get and let's see. So we have negative 1/16 2 10 and 1/2 is one time is another one. Half their 16 minus eight courses Aides. 16 squared times two so negative 1/16 times, 1/2 of getting negative 1 32nd close eight times two is 16 over 16. Squared is 1/16 and negative 1 32nd plus 1/16 is one over 32. So this one does not come out to zero, and we're done. Hopefully that helped.

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