Geometric Interpretation of the "Half Derivative"

Question

I've been doing a little bit of research into fractional calculus involving fractional derivatives, and I was wondering what the geometric interpretations of such derivatives would be.

As we know, the first derivative can represent slope or rate of change, and the second derivative can be used to investigate convexity or concavity. What about fractional derivatives, like the half-derivative? Is there any geometric interpretation or practical application for such a derivative?

Furthermore, finding such a derivative seems to be quite elusive. It is often tempting to find a "multiple derivative formula" such as $$\frac{d^n}{dx^n} \ln (x)=\frac{(-1)^{n-1}(n-1)!}{x^n}$$ and then evaluate it at fractional values, but this does not work. Because such a formula can be proven only by induction, evaluating it at fractional values is not justified.

Is there any way to find a fractional derivative without a formula proven by induction, or does a fractional derivative just have to be defined separately?

TLDR: What is the geometric interpretation of a fractional derivative? What are some applications? How do I find a fractional derivative?

Physical interpretations often come from fractional differential equations. — Simply Beautiful Art, Jul 15 '17 at 14:03
(Oh yeah, and on the side note, your false formula for the fractional derivatives of $\ln(x)$ fail to hold inductively backwards, that is, I can't plug in $n=0$ or any negative integers.) — Simply Beautiful Art, Jul 15 '17 at 14:38
@ Nilknart : For examples of applications in physics : Keith B.Oldham, Jerome Spanier, The Fractional Calculus, Academic Press, New York, 1974. In electrotechnics : page 11 in the paper "The fractional derivation": https://fr.scribd.com/doc/14686539/The-Fractional-Derivation-La-derivation-fractionnaire — JJacquelin, Jul 15 '17 at 17:20

Geometric Interpretation of the "Half Derivative"

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