a half-derivate or -integral or -fn.

as for "-- find f (x) so that f (f (x)) = exp (x) --"
,
find a halfDerivate[exp(x)] = exp^( '/2) [x] -- as for a^x = a^(x/2)·a^(x/2) the exp ' (x) = exp^( '/2) [x] "unknown operator" exp^( '/2) [x]

it's something like find half limit for Lim_(x→0) Sin(x) = halfLim_(x→0) Sin(x) "unknown operator" halfLim_(x→0) Sin(x)

(the target here is a "fraction space" in between "function space" and it's "differential space" so that the value 0.5 (a half) might be any fraction)
the speciffic target here is "fraction space" for fraction 0.5 and the function exp(x)

ci139 said:

as for "-- find f (x) so that f (f (x)) = exp (x) --"
,
find a halfDerivate[exp(x)] = exp^( '/2) [x] -- as for a^x = a^(x/2)·a^(x/2) the exp ' (x) = exp^( '/2) [x] "unknown operator" exp^( '/2) [x]

it's something like find half limit for Lim_(x→0) Sin(x) = halfLim_(x→0) Sin(x) "unknown operator" halfLim_(x→0) Sin(x)

(the target here is a "fraction space" in between "function space" and it's "differential space" so that the value 0.5 (a half) might be any fraction)
the speciffic target here is "fraction space" for fraction 0.5 and the function exp(x)

Click to expand...

Is there a question within your incoherent posting?

Ratch

What difference would that make?
Surprisingly your question contains a partial answer - as the wandering from function to it's derivate may turn out to be non-homogenous non-linear e.g. an incoherent travel - that also might not describe due limited scope and -vocabulary of today's math.