Partial derivatives and graphs

Hello everyone. So I have here the graph of a two variable function, and I'd like to talk about how you can interpret the partial derivative of that function.

So specifically, the function that you're looking at is f of x, y is equal to x squared times y plus sine of y. And the question is, if I take the partial derivative of this function, so maybe I'm looking at the partial derivative of f with respect to x, and let's say I want to do this at negative 1, 1. So we'll be looking at the partial derivative at a specific point. How do you interpret that on this whole graph?

So first, let's consider where the point negative 1, 1 is. If we're looking above, this is our x-axis, this is our y-axis. The point negative 1, 1 is sitting right there, so negative 1 move up 1, and it's the point that's sitting on the graph.

The first thing you might do is you say, well, when we're taking the partial derivative with respect to x, we're going to pretend that y is a constant. So let's actually just go ahead and evaluate that. So when you're doing this, it looks, it says x squared looks like a variable, y looks like a constant, sine of y also looks like a constant. So this is going to be, we do differentiate x squared, and that's 2 times x times y, which is like a constant, and then the derivative of a constant there is 0.

And we're evaluating this whole thing at x is equal to negative 1 and y is equal to 1. So when we actually plug that in, it'll be 2 times negative 1 multiplied by 1, which is negative 2. Excuse me, but what does that mean, right? We evaluate this and maybe you're thinking this is kind of a slight nudge in the x direction. This is the resulting nudge of f. What does that mean for the graph?

Well, first of all, treating y as a constant is basically like slicing the whole graph with a plane that represents a constant y value. So you know, this is the y-axis, and the plane that cuts it perpendicularly that represents a constant y value. This one represents the constant y value 1. But you could imagine it, you know, you could imagine sliding the plane back and forth, and that would represent various different y values.

So for the general partial derivative, you know, you can imagine whichever one you want, but this one is y equals 1, and I'll go ahead and slice the actual graph at that point and draw a red line. This red line is basically all the points on the graph where y is equal to 1. So I'll just kind of emphasize that, where y is equal to 1, and this is y is equal to 1.

So when we're looking at that, we can actually interpret the partial derivative as a slope because we're looking at the point here. We're asking how the function changes as we move in the x direction, and from single variable calculus, you might be familiar with thinking of that as the slope of a line. To be a little more concrete about this, I could say, you know, you're starting here, you consider some nudge over there, just some tiny step. I'm drawing it, you know, as a sizable one, but you imagine that as a really small step as your dx.

And then the distance to your function here, the change in the value of your function, as you said dx, but I should say partial x or del x partial f. And as that tiny nudge gets smaller and smaller, this change here is going to correspond with what the tangent line does. And that's why you have this rise over run feeling for the slope, and you look at that value, and the line itself looks like it has a slope of about negative 2.

So it should actually make sense that we get negative 2 over here, given what we're looking at. But let's do this. Let's do this with a partial derivative with respect to y. Let's erase what we've got going on here, and I'll go ahead and move the graph back to what it was, get rid of these guys.

So now we're no longer slicing with respect to y, but instead let's say we slice it with a constant x value. So this here is the x-axis, this plane represents the constant value x equals negative 1, and we could slice the graph there. I could kind of slice it, I'll draw the red line again that represents the curve, and this time that curve represents the value x or equals negative 1. It's all the points on the graph where x equals negative 1.

And now, when we take the partial derivative, we're going to interpret it as a slice, as the slope of this resulting curve. So that slope ends up looking like this—that's our blue line. And let's go ahead and evaluate the partial derivative of f with respect to y.

So I'll go over here, use a different color. So the partial derivative of f with respect to y, partial y. So we go up here, and it says, okay, I see x squared times y. It's considering x squared to be a constant now, so it looks, it then says x, you're a constant, y, you're the variable. Constant times the variable, the derivative is just equal to that constant—so that x squared.

And over here, sine of y, the derivative of that with respect to y is cosine y. Cosine y. And if we actually want to evaluate this at our point negative 1, 1, what we'd get is negative 1 squared plus cosine of 1. And I'm not sure what the cosine of 1 is, but it's something a little bit positive.

And the ultimate result that we see here is going to be, you know, 1 plus something. I don't know what it is, but it's something positive, and that should make sense because we look at the slope here and it's a little bit more than one. Not sure exactly, but it's a little bit more than one.

So you'll often hear about people talking about the partial derivative as being the slope of the slice of a graph, which is great if you're looking at a function that has a two variable input and a one variable output, so that we can think about its graph.

And in other contexts, that might not be the case. Maybe it's something that has a multi-dimensional output. We'll talk about that later when you have a vector-valued function, what its partial derivative looks like. But maybe it's also something that has, you know, a hundred inputs, and you certainly can't visualize the graph.

But the general idea of saying, well, if you take a tiny step in a direction, here I'll actually walk through it in this graph context again. Now you're looking at your point here, and you say we're going to take a tiny step in the y direction, and I'll call that partial y, and you say that makes some kind of change. It causes a change in the function, which you'll call partial f, and as you imagine this getting really, really small and the resulting change also getting really small, the rise over run of that is going to give you the slope of the tangent line.

So this is just one way of interpreting that ratio, the change in the output that corresponds to a little nudge in the input. But later on, we'll talk about different ways that you can do that.

So I think graphs are very useful; when I move that, the text doesn't move. I think graphs are very useful for thinking about these things, but they're not the only way, and I don't want you to get too attached to graphs, even though they can be handy in the context of two variable input, one variable output. See you next video.