Let the $H$ be a collection of real valued invertible functions, define $f\circ g$ as composition, $f+g$ as the function $f+g(x):=f(x)+g(x)$ and define a family of functions $\{D_h\}_{h\in \Bbb R}:H\rightarrow H$
$D_h[f]:={f(x+h)-f(x)\over h}$
so we define the family of functions $\{F_h\}_{h\in \Bbb R}:H\rightarrow H$
$F_h[f]:=D_h[f]\circ f^{-1}$
define the powers of $F_h[f]$ as the iteration $$F^0_h[f]:=f$$ $$F^{n+1}_h[f]=F_h[F^{n}_h[f]]$$
Let's assume that this definition has not problems and that for a well behaved class of functions $H$ the family $F_h$ is well defined, or at least for some indexes $h$, and that the function is iterable.
the question is
Q1- Is possible to express the powers $F^{n}_h[f]$ in terms of the powers of $D^{n}_h[f]$? There is a closed form like for example for the powers of the difference operator or the higher order differentiation?
Q2- If in $D_h$ $h$ goes to $0$ we should have the derivative of $f$, then $D[f]\circ f^{-1}$ has some special properties? is it already known/useful somewhere?
eg. $\displaystyle\Delta_h[f](x)=\sum_{i=0}^n(-1)^i\binom{n}{i}f(x+(n-i)h)$ or higher order differentiation
Note: I understand that it could be the case that $D_h[f]$ is a costant function so if $D_h$ is a function on $H$ then $H$ could have non-invertible element and this wold make $F_h$ not defined on all the domain $H$.
Note: If $\lim_{h\to 0^+}D_h$ then $D[f]=f$. In this case for $\lim_{h\to 0^+}F_h=F$ we have the following
$$F[f]\circ f=f'$$
So $F[f]$ is the solution of the following functional equation $\chi\circ f=f'$ OR $$\chi(f(x))=f'(x)$$