Welcome to course three, module four, this is the last module of course three. And really if you're taking the whole specialization, this is the last module of the entire thing. So the last concepts we're going to tackle in this module, are partial derivatives. And then we'll get to things like directional vectors, and gradients. So first let's start off with partial derivatives, let's talk about a little bit, and talk a little bit about what's going on here. So in general we're only dealing with right now, at least for this two example, a f(x,y). So the general setup is you have a function of multiple variables in this case it's x, y, so it's two variables. You can have three, you can have four, you can have as many as you want, depending on the dimension of the function that you're dealing with. In this case we just have x and y, we have Z and we've set it equal to it just for notation. Okay, so for example, Z might be x squared, y- y squared, square root of x + x. I just made that up, it doesn't matter, it's just an example of what this might be. So it's just a function of two variables. And in general the idea of partial derivatives, is that generally when you have a function of one variable. Let's say you have x squared, when you take the derivative, it gives you the rate of change at a point, right? That's the entire point of taking the derivative, or at least one of the major reasons, why you might do it. So for partial derivatives, the idea is that you want to take the derivative, but you're only going to get a partial rate of change. So you might say, okay, I have a function that deals with two variables. Now, I want to see at a given point, how x is changing. The idea is that if, you want to know, how x is changing in a function with two variables that have x and y, then y it needs to stay the same, right? y needs to stay constant throughout this entire process? And then you move x around and you see, how the rate of change changes. So for partial derivatives, the concept is that with a function of many variables, if you want the rate of change of 1, you need to keep the rest of them constant. Okay, so what you do is, you take the derivative of the function in terms of the variable you want to change, and you treat the rest of them as constant. So in this case we have f (x,y), and we're taking this is the partial derivative of the function in terms of x. This is just a bunch of notation. Right, this is a lot of different ways that when, you hit data science or you hit more advanced math in certain ways. This is the notation that you might see, and when you do, you'll know it, they're talking about a partial derivative of the function. And so this is saying a f (x,y), I'm taking the partial derivative in terms of x. You can also think about it, with this notation, again, with all these notations. So this one is from personal experience, this one isn't crazy common. This one you will see, this is probably the most common, these two are probably the most common here. But you might see Z, has been defined as a f (x,y), so you might see the derivative of Z, and then the subscript x. So, it's saying the derivative of Zx, and you might see something like this. But in general, I'd say overwhelmingly from personal experience that these two with, maybe a little bit of this one is, what you'll mainly see. So when you see this, you should know it's partial derivatives, and it's taking the derivative of a function of multiple variables. Keeping all of them constant except, the one that you want to take, the derivative in terms of. And next time we'll do plenty of examples, and we'll get really comfortable with this going forward.
