The "little" differential operator and applications to tetration

[JmsNxn]

returning to standard notation and at first glance , it seems the function you search for is any one ( coo ) that satisfies

f(x) = exp(f(x))

that function has a " history " here.

... at first glance ... im in a hurry ...

I'm not really sure how you got there?

Basically, I'm looking for a function that will have a little derivative that is translated from the tetration derivative. By "translated" I mean:

\( 1:\frac{d}{dx} xn = x(n-1) + ln(n) \) is a translation of \( \frac{d}{dx} x^n = nx^{n-1} \) and \( 1:\frac{d}{dx} x + y = y \) is a translation of \( \frac{d}{dx} xy = y \).

So I'm looking for a function that will have a translated little derivative of sexp, so that we can derive the derivative of sexp. So basically we first have to design a law as to how translated derivatives are related, and then try to derive one for tetration. My first instinct was that it would be exponentiation, but I doubt that now.

I'm still a little sketchy now but it's coming to me slowly, I'm sure you know that feeling.