Gradient and directional derivatives
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- [Voiceover] So here I'm gonna talk about the gradient. And in this video, I'm only gonna describe how you compute the gradient, and in the next couple ones I'm gonna give the geometric interpretation. And I hate doing this, I hate showing the computation before the geometric intuition since usually it should go the other way around, but the gradient is one of those weird things where the way that you compute it actually seems kind of unrelated to the intuition and you'll see that. We'll connect them in the next few videos. But to do that, we need to know what both of them actually are. So on the computation side of things, let's say you have some sort of function. And I'm just gonna make it a two-variable function. And let's say it's f of x, y, equals x-squared sine of y. The gradient is a way of packing together all the partial derivative information of a function. So let's just start by computing the partial derivatives of this guy. So partial of f with respect to x is equal to, so we look at this and we consider x the variable and y the constant. Well in that case sine of y is also a constant. As far as x is concerned, the derivative of x is 2x so we see that this will be 2x times that constant sine of y, sine of y. Whereas the partial derivative with respect to y. Now we look up here and we say x is considered a constant so x-squared is also considered a constant so this is just a constant times sine of y, so that's gonna equal that same constant times the cosine of y, which is the derivative of sine. So now what the gradient does is it just puts both of these together in a vector. And specifically, maybe I'll change colors here, you denote it with a little upside-down triangle. The name of that symbol is nabla, but you often just pronounce it del, you'd say del f or gradient of f. And what this equals is a vector that has those two partial derivatives in it. So the first one is the partial derivative with respect to x, to x times sine of y. And the bottom one, partial derivative with respect to y X-squared cosine of y. And notice, maybe I should emphasize, this is actually a vector-valued function. So maybe I'll give it a little bit more room here and emphasize that it's got an x and a y. This is a function that takes in a point in two-dimensional space and outputs a two-dimensional vector. So you could also imagine doing this with three different variables. Then you would have three partial derivatives, and a three-dimensional output. And the way you might write this more generally is we could go down here and say the gradient of any function is equal to a vector with its partial derivatives. Partial of f with respect to x, and partial of f with respect to y. And in some sense, we call these partial derivatives. I like to think as the gradient as the full derivative cuz it kind of captures all of the information that you need. So a very helpful mnemonic device with the gradient is to think about this triangle, this nabla symbol as being a vector full of partial derivative operators. And by operator, I just mean like partial with respect to x, something where you could give it a function, and it gives you another function. So you give this guy the function f and it gives you this expression, this multi-variable function as a result. So the nabla symbol is this vector full of different partial derivative operators. And in this case it might just be two of them, and this is kind of a weird thing because it's like what, this is a vector, it's got like operators in it, that's not what I thought vectors do. But you can kind of see where it's going. It's really just, you can think of it as a memory trick, but in some sense it's a little bit deeper than that. And really when you take this triangle and you say ok let's take this triangle and you can kind of imagine multiplying it by f, really it's like an operator taking in this function and it's gonna give you another function. It's like you take this triangle and you put an f in front of it, and you can imagine, like this part gets multipled, quote unquote multiplied with f, this part gets quote unquote multiplied with f but really you're just saying you take the partial derivative with respect to x and then with y, and on and on. And the reason for doing this, this symbol comes up a lot in other contexts. There are two other operators that you're gonna learn about called the divergence and the curl. We'll get to those later, all in due time. But it's useful to think about this vector-ish thing of partial derivatives. And I mean one weird thing about it, you could say ok so this nabla symbol is a vector of partial derivative operators. What's its dimension? And it's like how many dimensions do you got? Because if you had a three-dimensional function that would mean that you should treat this like it's got three different operators as part of it. And you know I'd kinda, finish this off down here, and if you had something that was 100-dimensional it would have 100 different operators in it and that's fine. It's really just again, kind of a memory trick. So with that, that's how you compute the gradient. Not too much too it, it's pretty much just partial derivatives, but you smack em into a vector where it gets fun and where it gets interesting is with the geometric interpretation. I'll get to that in the next couple videos. It's also a super important tool for something called the directional derivative. So you've got a lot of fun stuff ahead.