Finding local iterational derivatives and integrals

By analogy with discrete multiplicative derivative of a function $f(x+1)\frac1{f(x)}$ we can define a discrete iterative derivative $f(x+1)\circ f^{[-1]}(x)$. The inverse operator to it (discrete iterative integral) is known as superfunction or flow. It exhibits many properties common to integral such as that it is more difficult to express in closed form than the derivative and that a constant parameter appears in the process.

But what if to consider local iterative integrals and derivatives? Lets define the local iterative derivative as follows:

$$f^{\circ}(x)=\lim_{h\to\infty} \left( f \left(x+\frac xh\right)\circ f^{[-1]}(x)\right)^{[h]}$$

where in the square brackets is the number of iterations. And also by analogy we can define the local iterative integral as an inverse operator.

Any thoughts how to find such derivatives and integrals and possibly some more examples?

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