Wow! From the article:

(And pi turns up in an unlikely spot again)

Which shows that (in general) if you can think of extending an idea in mathematics, you probably can! What's next ... complex powers?

I've been thinking quite a bit about this (off and on) since it was first posted. When I first read the wiki, it was very much over my head, but now I think I get it. Some further thoughts (please correct me if I'm wrong):

Seeing how:

and it follows that:

for any whole number "n" is it fair to assume:

I think so. But that's easy. For my next trick I'll be using the trig identities:

I think you all probably already see where this is going:

And if "n" is expanded into all real numbers:

Is my logic sound? Does anyone have anymore half derivatives to add?

On a slightly unrelated note, what's the deal with the notation? I've never understood it... why is the second derivative:

and not

?

Ok, that makes a bit more sense, I never treated "dx" as a single entity, I always thought it was d(x²) not (dx)²

What about my other work? Any good or am I a little off?