Evaluate: $$\Large \dfrac{\mathrm d^{1/2}}{\mathrm dx^{1/2}}~\huge x$$

Let's try to solve this without relying outside sources. :)

Question

Evaluate:   $$\Large \dfrac{\mathrm d^{1/2}}{\mathrm dx^{1/2}}~\huge x$$

Let's try to solve this without relying outside sources. :)

anonymous · Answer

A fractional order derivative? O_o Thats new to me, haha.

geerky42 · Answer

Yeah haha.

This sort of thing, is from "Fractional Calculus" branch and it is still wide open.

geerky42 · Answer

You can look it up. At my glance, it is ugly and quite hard to grasp.

anonymous · Answer

Is this something you just randomly came across or is it something you need to know? xD

geerky42 · Answer

Randomly came accross, yeah lol.

Feeling like raising awareness to this branch a little bit. And I like to see users go like "wtf mind-blown," you know? haha

anonymous · Answer

Hm. So what about it is messing with ya?

geerky42 · Answer

Actually I was hoping to see how different users would approach to this, but seem I did it in bad time since not many users came...

Anyway I was thinking exactly how to interpret half derivative. Maybe we can find a way to like find "average" of $\dfrac{\mathrm d^n}{\mathrm dx^n}$ and $\dfrac{\mathrm d^{n-1}}{\mathrm dx^{n-1}}$

Of course, I have completely no idea where to go from here.

anonymous · Answer

Well, defining the derivative in a general way makes sense. Looking at the wiki article, finding a way to generalize the derivative as a formula and then plug in values makes sense (since Ive seen the concept in how the gamme function generalizes factorials) and seems like itd be the only way of doing it. But yeah, the interpretation is an odd idea. I guess applying it to any possible equations. I dont know any equations off the top of my head, but any value or result that is dependent on the order of the derivative Im sure could be an application. Maybe PDEs can make use of it? In a physical context, your idea of an average sounds like a good possibility.