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# Directional derivative, formal definition

## Video transcript

so I have written here the formal definition for the partial derivative of a two variable function with respect to X and 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 the little thing on top this will be some vector in the input space and 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 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 and when you're taking the partial derivative at a point a B you're looking over here and you say maybe that's your point some point a B and you imagine nudging it slightly in the x-direction and saying hey how does that how does that influence the function so maybe this is where a B lands and maybe the result is a nudge that's a little bit negative that would be a negative partial derivative and 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 you 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's like nudge and you 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 B I'm going to say it's a you know just a and then make it clear that that's a vector and this will be a two dimensional vector so put that little little arrow on top to indicate that it's a vector and if we rewrite this definition we'd be thinking the limit as H goes to zero of something divided by H but that's saying 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 and they 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 and it's common to use you know this little eye 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 and 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 and 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 this vector here what you multiply your nudge by as you're adding the adding the input so let's just let's just rewrite that over here in the context of 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 you know 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 and 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 initial input vector plus H that scaling value that little nudge of a value multiplied by the vector whose Direction we care about and then you subtract off the value of F at that original 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 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 a nudge purely in the x-direction no erase these guys you would think at this point as being a 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 and then H times V you know maybe V is some some vector often you know 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 what's the resulting nudge to the output and the ratio between the size of that resulting nudge to the output and the original guy there is your directional derivative and more importantly as you take the limit for that original nudge getting really really small that's going to be your directional derivative and 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 2 you know if you go over here and you start plugging in 2 times V and seeing how that influences things it'll be twice the twice the change because here even if you're scaling by the same value H it's going to double the double the initial nudge that you have and it's going to double the resulting nudge out here even though the denominator H doesn't stay change so when you're taking the ratio what you're considering is the the size of your initial nudge actually might might be influenced so 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 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 Y that should double the size of your of your derivative but I'll get to that in following videos this right here is is the formal definition to be thinking about and I'll see you next video