00:01
Okay, so for this problem, we are given that we have a function f of x, y, and we want to find the maximum rate of change and the direction in which it occurs at 0 .2 comma 4.
00:13
Now, if we want to find that if the direction at which the maximum rate of change occurs, that always corresponds to the gradient.
00:22
So the max rate of change occurs in the direction of the gradient.
00:30
And the magnitude of this rate of change is just the magnitude of the gradient.
00:38
And so all we have to do is calculate the gradient of this function at the specified point.
00:44
And then we compute the magnitude.
00:47
So the direction here is given by the gradient, which is f sub x and f sub y.
00:53
Just the vector form of f sub x and f sub y.
00:57
So the derivative of this function with respect to x, is going to be, well, as a reminder, we can maybe rewrite this as y squared x to the negative first.
01:06
If you're struggling like that, you move down the negative 1.
01:10
And we get as our derivative, negative y squared over x squared.
01:17
So we would get in this first part, it would be negative y squared over x squared.
01:21
And this part right here, the y component is just going to be 2y over x.
01:27
Now plugging in my value for x and y, for 2 and 4, we get negative 4 squared over 2 squared.
01:39
We get 2 times 4 divided by 2, which in this case is going to be negative 16 over 4.
01:48
This one is going to just be 4.
01:50
And so we get negative 4 4, which means just kind of looking at my graph here, if i had a graph, and i was at my point 2 comma 4, which is right up here, this is saying that if i move in the direction of negative 4, 4, which is kind of like in this direction here.
02:06
I'm moving this direction...