How to define the rate of change function that is "between " a function f and its derivative function f ' ( In case such a thing exists)

Question

[ There is an apparently similar question involving the concept of " linear operator". However, my question is asked at a much more basic level. I even do not know what is a "linear operator". Please, do not label my question as " duplicate"]

To calculate f', we use a certain formula, more precisely a ratio

f(x+h) - f(x) / (x+h) -x.

I'd like to decompose the process that leads from f to f ' .

If I wanted to define this ratio as a function that would take place " between" f and f ', what would be the domain of this intermediary function ? Would it be a function of 2 variables : x and h?

An attempt.

Let f be a function.

I define a function g.

In order to define the domain of g, I first define a set

H = {h | there exists x2 and x1 belonging to Dom(f) such that h= x2 - x1 }

I then define : Dom (g) = the cartesain product of Dom(f) and of H.

Formula for g : g( x,h) = f(x+h)-f(x)/ (x+h)- x

And finally : f ' (x) = lim ( as h tends to 0) of g(x,h)

Domain of $g$ is ${(x,h): h \neq 0, x\in Dom(f),x+h \in Dom(f)}$. — Kavi Rama Murthy, Apr 13 '19 at 11:39
You might find that this Wikipedia article on fractional calculus answers your question. It gives a meaning to $\dfrac{d^\frac12y}{dx^\frac12}$, among other things — TonyK, Apr 13 '19 at 11:43
@Kavi Rama Murthy. Thanks. Simpler and more logical indeed ! — , Apr 13 '19 at 11:45
@MarkS. Thanks for your remark. I've edited my question in order to take your remark into account. — , Apr 13 '19 at 12:00
@EleonoreSaintJames Sorry, I don't understand the term platonistic. — Kavi Rama Murthy, Apr 13 '19 at 12:17
@ KaviRamaMurthy. I should rather have said " platonist" , I meant that the quotation expresses an idea pertaining to Plato's philosophy ( but sure, Plato is not the only one who has expressed such a true idea). — , Apr 13 '19 at 12:21
@TonyK, maybe expand that into an answer? Eleonore, even if you are not familiar with the way that question is phrased, your question at least has the same answer as that other one. — Mark S., Apr 13 '19 at 12:49
@MarkS, The OP is asking about how to define the function $$g(x,h) = \frac {f(x+h) - f(x)}{(x + h) - x}$$ in particular, its domain. And it is Kavi Rama Murthy who has answered this, not TonyK. This question has nothing to do with other concepts of differentiation. — Paul Sinclair, Apr 13 '19 at 17:52

How to define the rate of change function that is "between " a function f and its derivative function f ' ( In case such a thing exists)

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