I'm new to this kind of stuff so maybe this is a stupid question but I don't even know what to search on the internet.
My problem is that: find the derivative of the following function on $\Bbb R^3$
$g(x;y)=D^xf(y)$
Where $D^x$ is the x-th derivative of the function $f(y)$ defined on $\Bbb R^2$ (assuming that is continuos and always differentiable). I know that, using fractional calculus, $D^x$ is well defined for every $x\in \Bbb R$ but I don't know if the function I wrote has any sense or which branch of math studies this kind of relations so I don't even know how to start. Any hint ?
Just to make things clearer I make an example:
Take $f(y)=y^2$ as the function, some values would be:
$g(1;y)=2y$
$g(2;y)=2$
$g(3;y)=0$
The derivative of a function on $\mathbb R^3$ is not really a good way to put it. You can take the partial derivative (which is the derivative with respect to a specific variable). In addition, if you have a function defined on $\mathbb R^2$ I don't think trying to find derivatives in $\mathbb R^3$ is going to make much sense (fractional calculus aside).
Having said all of that, if you have a function $$F=f(y)$$ you can use the derivative operator to find z given... $$(1) \quad z=D^x \cdot f(y)$$
To answer your question, equations like $(1)$ are studied all the time in mathematics. If you have $z$, but would like $f(y)$ you need integrals and possibly the theory of differential equations. If you have $f(y)$, but would like z, you need the theory of differentiation. If your interested in non-integer derivatives then replace the above subjects with the prefix-"fractional".
Here are some links, Intro to Fractional Calculus,
Intro to Fractional Differential Equations, Interpretation of Fractional Derivative, Geometric Interpretation of Fractional Integration. The last link I provided is a bit silly, as it just is a paper about vector projection, it doesn't actually provide physical intuition, it just identifies how one would plot the integral.