Algebraic derivatives of a given function

Question

I was studying about fractional derivatives and after that did some own work on imaginary order derivatives. Now I'm curious to know are algebraic order derivatives possible? Like the xth derivative of x.

Answer 1

Continuing my comment:

You should notice how this is an entirely new structure. As when we differentiate we usually do it a $\mathbb{N_+}$ - [integer] amount of times. Now people generalized this idea to the $\mathbb{Q}$ - [rational numbers], $\mathbb{R}$ - [reals] or even $\mathbb{C}$ - [complex numbers]. The common idea is numbers. But the core concept stems from a discrete amount of operations.

But [x] is not a number. It can be a function or an indexing letter or a parameter. Now when you differentiate a function $f(x)$ you don't just take it at one point (like $x_0$) . You try to find its derivative on an as big domain as possible (often $\mathbb{R}$ and its varieties).

If you want to to differentiate $[x]$ amount of times while $x$ is going from $-\infty$ to $\infty$ (so the domain being $\mathbb{R}$ presumably) then you'll have to find a different kind of derivative for every single point and you'll run into some problems.

Using the original idea, you'd need to calculate a lot.

Namely $$\text{card}(\mathbb{R})=\mathfrak{c}=2^{\aleph_0}$$

amount of times. You can imagine $\aleph_0$ as how many rationals are there. So you'd need to calculate an exponential amount.

Of course this logic is rather the computer science side of the problem. You could just design a new structure where the derivative-operator continously changes as we shift through the reals. But figuring out that is rather your job.

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