So like, a functional derivative of differentiation? Except like, instead of the 2nd order function returning a number, where the functional derivative would b

So like, a functional derivative of differentiation? Except like, instead of the 2nd order function returning a number, where the functional derivative would be ((R -> R) -> R) -> ((R -> R) -> R), uh, instead, the thing you got. Ok, but, differentiation is linear, so, (\frac{d}{dx}(u(x) + \varepsilon \eta(x)) - \frac{d}{dx}(u(x)) )/\varepsilon is just \frac{d}{dx}(\eta(x)) (and so there's not even much need to take the limit as \varepsilon goes to 0, as it doesn't depend on \varepsilon anyway) .. Uh, that seems a little odd, that the derivative of differentiation in the direction of a function would be the derivative of that function? Like, the derivative of something linear should be a constant, ah, but, the directional derivative of 5 x_1 + 3 x_2 also depends on the direction in which it is taken, though it doesn't depend at all on where it is taken. Ok, yeah, same situation here then. Is that the sort of thing you are talking about? Though, you said "of a derivative" not "of the derivative", so I guess maybe you mean like, more generally things that satisfy Liebniz's law?