3

There are many notations for a derivative of $y$ with respect to $x$. Two, most popular are $y'(x)$ or just $y'$ and $\frac{dy}{dx}$.

For higher order derivatives, the more consistent notation is $\frac{d^ny}{dx^n}$.

Now, we know it is possible to have fractional derivative orders (there was even one question about it here on math.stackexchange). What about irrational derivative orders?

For example, does this expression exist?: $$\frac{d^{\sqrt{2}}}{dx^{\sqrt{2}}}\Bigg(2x^3+5x\Bigg)$$

KKZiomek
  • 4,012
  • 6
    Yes, what is called "fractional derivative" actually applies here, it's not limited to derivatives of rational order, you can take it to be any arbitrary complex number. – Count Iblis Apr 24 '16 at 19:53
  • 1
    Do you have a use for derivatives with irrational orders? We could get curious about any fanciful combination of mathematical objects and symbols, after all. –  Apr 24 '16 at 20:33
  • See this and this. –  May 08 '16 at 06:51

2 Answers2

2

Short answer: Yes.

The name "fractional calculus" is an unfortunate one, because it suggests that the theory only handles rational orders. But, as Iblis mentions in the comments, the theory also deals with irrational and even complex orders.

Eric Stucky
  • 12,758
  • 3
  • 38
  • 69
0

See my reply to Is there a notion of a complex derivative or complex integral? for an example of a complex differentiation resulting in plane curve.

Community
  • 1
Cye Waldman
  • 7,524