Directional derivative, formal definition

So I have written here the formal definition for the partial derivative of a two-variable function with respect to X. What I want to do is build up to the formal definition of the directional derivative of that same function in the direction of some vector v, and you know, V with a little thing on top. This will be some vector in the input space.

Um, I have another video on the formal definition of the partial derivative if you want to check that out. And just to really quickly go through here, I’ve drawn this diagram before, but it’s worth drawing again. You think of your input space, which is the XY plane, and you think of it, you know, somehow mapping over to the real number line, which is where your output F lives.

When you're taking the partial derivative at a point AB, you're looking over here and you say maybe that's your point, some point AB, and you imagine nudging it slightly in the X direction and saying, "Hey, how does that influence the function?" So maybe this is where AB lands, and maybe the result is a nudge that's a little bit negative; that would be a negative partial derivative. You think of the size of that nudge as partial X, and the size of the resulting nudge in the output space as partial F.

So the way that you read this formal definition is you think of this variable H. You know, people could say Delta X, but H seems to be the common variable people use. You think of it as that change in your input space, that slight nudge, and you look at how that influences the function when you only change the X component here. You know, you're only changing the X component with that nudge, and you say, "What's the change in F? What's that partial f?"

So I'm going to write this in a slightly different way using vector notation instead. I'm going to say, you know, partial F partial X, and instead of saying the input is A, I'm going to say it's, you know, just A, and then make it clear that that's a vector. This will be a two-dimensional vector, so I'll put that little arrow on top to indicate that it's a vector.

If we rewrite this definition, we'd be thinking the limit as H goes to zero of something divided by H, but that thing now that we're writing in terms of vector notation is going to be F of... So it's going to be our original starting point A, but plus... what I mean up here, it was clear we could just add it to the first component. But if I'm not writing in terms of components, I have to think in terms of vector addition. Really, what I'm adding is that H times the vector, the unit vector in the X direction.

It's common to use, you know, this little I with a hat to represent the unit vector in the X direction. So when I'm adding these, it's really the same, you know. This H is only going to go to that first component, and the second component is multiplied by zero. What we subtract off is the value of the function at that original input, that original two-dimensional input that I'm just thinking of as a vector here.

When I write it like this, it's actually much clearer how we might extend this idea to moving in different directions, because now all of the information about what direction you're moving is captured with this vector here. Um, what you multiply your nudge by as you're adding the input.

So let's just rewrite that over here in the context of the directional derivative. What you would say is that the directional derivative in the direction of some vector, any vector, of f evaluated at a point, and we'll think about that input point as being a vector itself A. Here, I'll get rid of this guy. It's also going to be a limit, and as always with these things, we think of some... not, I mean, always, but with derivatives, you think of some variable as going to zero.

Then that's going to be on the denominator, and the change in the function that we're looking for is going to be F evaluated at that initial input vector plus h, that scaling value, that little nudge of a value multiplied by the vector whose direction we care about. Then, you subtract off the value of F at that original input.

So this right here is the formal definition for the directional derivative, and you see how it's much easier to write in vector notation because you're thinking of your input as a vector and your output as just some nudge by something.

So let's take a look at what that would feel like over here. You know, instead of thinking of DX and a nudge purely in the X direction... no, erase these guys. You would think of this point as being a vector, vector-valued A. So just to make clear how it's a vector, you'd be thinking of, you know, it's starting at the origin, and the tip represents that point.

Then H * V, you know, maybe V is some, uh, some vector off in, you know, a direction that's neither purely X nor purely Y. But when you scale it down, you know, you scale it down, it'll just be a tiny little nudge. You know, that's going to be H, that tiny little value scaling your vector V. So that tiny little nudge, and what you wonder is, "Hey, what's the resulting nudge to the output?" The ratio between the size of that resulting nudge to the output and the original guy there is your directional derivative.

More importantly, as you take the limit for that original nudge getting really, really small, that's going to be your directional derivative. You can probably anticipate there's a way to interpret this as the slope of a graph. That's what I'm going to talk about next video.

But you actually have to be a little bit careful because we call this the directional derivative. But notice if you scale the value V by two, you know, if you go over here and you start plugging in 2 * V and seeing how that influences things, it'll be twice the change. Because here, even if you're scaling by the same value H, it's going to double the initial nudge that you have, and that's going to double the resulting nudge out here, even though the denominator H doesn't stay changed.

So when you’re taking the ratio, what you're considering as the size of your initial nudge actually might be influenced. Some authors, they’ll actually change this definition, and they’ll throw a little, you know, absolute value of the original vector just to make sure that when you scale it by something else it doesn’t influence things, and you only care about the direction.

But I actually don’t like that. I think there's some usefulness in the definition as it is right here, and that there’s kind of a good interpretation to be had for when if you double the size of your vector, why that should double the size of your derivative.

Um, but I'll get to that in following videos. This right here is the formal definition to be thinking about, and I'll see you next video.