I know this question sounds strange and I'm not exactly sure how to explain however I will try my best. I do not think this question will have any mathematical uses; however, personally I find it interesting. I would like to hear everyone's thoughts on this.
First the question is based in fraction calculus so the n in $\frac{d^n}{dx^n}$ does not have to be a natural number. meaning things like semi derivatives exist and the overall calculations are difficult. Luckily there are formulas for the general fractional derivative of set functions.
However, what would be the nature of a curve based on fractional derivatives and being evaluated at that point? The best way to illustrate my question is probably by showing an example for $f(c) = x^2$.
The general equation for a fractional derivative of $f(x) = x^k$ is: $$ \begin{equation} \frac{d^nf(x)}{dx^n} x^k = \frac{\Gamma(k+1)}{\Gamma(k+1-n)} x^{k-n} \end{equation} $$ Therefore, taking steps of dx = 0.1 for $0\leqslant x \leqslant 2$ would result in: $$ \begin{array}{c|c||c} x & \frac{d^xf}{dx^x} & \frac{d^xf}{dx^x}\\ \hline 0 & \frac{\Gamma(3)}{\Gamma(3-0)} x^{2-0}=x^2 & 0 \\ 0.1 & \frac{\Gamma(3)}{\Gamma(3-0.1)} x^{2-0.1}\approx1.09448 x^{1.9} & 0.0138 \\ 0.2 & \frac{\Gamma(3)}{\Gamma(3-0.2)} x^{2-0.2}\approx 1.19297 x^1.8 & 0.0658 \\ \vdots & \vdots & \vdots \\ 1 & 2x & 2 \\ \vdots & \vdots & \vdots \\ 2 & 2 & 2 \\ \end{array} $$ Which if plotted on a graph looks like this:
Unfortunately, my maths skills nor knowledge are not good enough to answer my question as to what is the nature of such a curve or surface if in the complex domain. Therefore, I wanted to have input from other people about this question. Of course this can be done for a function like $\sin(x)$ or the derivative instead of being to the order of x could be to the order of a function g(x). Any ideas how I should go about analysing these graphs and find their nature or properties for these and other functions?
Thank you in advance!