00:01
In this question, we're given z as a function of w, w as a function of x and y, and x and y are h functions of r and s.
00:09
And we're asked to solve for the partial derivative of z with respect to r at a particular value for rns.
00:19
So let's start with a dependency diagram.
00:26
So we start with z, which is a function of w, so which means this is w.
00:36
And w is a function of x and y.
00:38
So from w, we're going to branch off into x and y.
00:46
And then for this part, they converge to r.
00:56
Okay, so set to w, this would be d zddd w.
01:04
W to x would be partial w, partial w, partial x, and partial w, partial and this would be partial x partial r okay so this means if we want to find the partial derivative of z with respect to r we have to go down both of these branches so this means it's going to be equal to b z d w partial w partial x partial x partial x partial x partial x partial x partial r plus dzdw, partial w, partial y, and then partial y, partial y, partial r.
01:59
Okay, so now that we have that, what we want is if we want the partial derivative z with respect to r at r equals 1 and s equals 0.
02:14
Okay, so what we're going to do is we're going to take these two values and look at our diagram and go backwards.
02:21
Okay, so if we go backwards, we go backwards, we'll do.
02:23
We know that when r is when s is zero, we can just plug into our equation here to find x and the equation here to find y.
02:33
Okay.
02:35
So this means our x is going to be equal to two times one to the power of three minus zero.
02:47
So that is going to be two.
02:49
And then our y is going to be one times e to the zero and that's going to be one.
02:55
So if we go backwards again to find w, this is saying w is g of xy.
03:03
So this means w is going to be g of 2 comma 1.
03:08
And we're actually given that from the question right here.
03:12
That is equal to 7.
03:16
So now that we have these values, we can actually look back here, and then we can plug those values in.
03:27
So this means we're going to need to have the value for we're going to need dz, d, w, and we want that evaluated at w equals 7.
03:41
So this is going to be at w equals 7.
03:44
And it's going to be times partial w, partial x.
03:48
And this is going to be evaluated at x equals 2, y equals 1.
03:54
And last one is going to be partial x, partial r.
03:58
And that's going to be at r equals 1 s equal 0 and it's going to be plus and this is going to be again same thing d zddd at w equals 7 partial w partial w partial y at x equals 2 y equals 1 and lastly partial y partial r at r equals 1 s equal 0 okay so we have most of these values the only ones that we don't have are the partial x partial r and the partial y partial r because those we need to actually take the partial derivatives ourselves so let's do that so we'll do that on the side actually we can do it under here so we have partial x partial r that is going to be equal to six r squared so which means if we want to evaluate it at the values are equals 1 s equals 0, this is going to be equal to 6.
05:10
And same thing, partial y partial r, that is going to be e to the s, which means if we want to evaluate it at r equals 1 and s equals 0, this is going to be 1.
05:25
So now that we have these two values, we can plug everything in.
05:29
So this is going to be equal to.
05:32
This is dzdw.
05:36
Z is a function w, so dzdw is just really f prime.
05:40
So f prime of 7 is negative 1.
05:42
We're given that partial w, partial x, that is, that w is a function g...
