The "little" differential operator and applications to tetration

[tommy1729]

I've had another little thought bubble, this time it involves the geometric derivative.

if
\( -1:\frac{d}{dx}f(x) = \lim_{h\to\0}(\frac{f(x+h)}{f(x)})^{\frac{1}{h}} \)
is the geometric derivative, abbreviated as:
\( -1:\frac{d}{dx}f(x) = e^{\frac{f'(x)}{f(x)}} \)

then:
\( -1:\frac{d}{dx} sexp_e(x) = e^{\frac{d}{dx}sexp_e(x-1)} \)

Now what I am looking for are two functions that behave the same for the little derivative and the little geometric derivative. I believe that one of these functions will have a little derivative which will essentially "look the same" as the tetration derivative, we'll only have to raise the operators by one and remove lns where applicable. I'm not sure how rigorous this is all sounding now, but I'll try to be as rigorous as possible when writing it out mathematically.


\( 1:\frac{d*}{dx} f(x) = \lim_{h\to\0}\,\frac{f(x\, \{-1\}\, ln(h)) - f(x)}{h} \) is the little geometric derivative.

closest abbreviation is
\( 1:\frac{d*}{dx} f(x) = \lim_{h\to\0}\,\frac{f(ln(e^x+h)) - f(x)}{h} \)

\( 1:\frac{d*}{dx} e^x = 1 \)
but
\( 1:\frac{d*}{dx} e^{e^x} = e^{e^x} \) just like the geometric derivative.

The fact that they both coincide there leaves me optimistic. This is leading me somewhere but I have to work out some more on paper first.

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 ...