for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom f & f = ln * (exp_R / ((#Z 2) * (f1 + exp_R))) & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = (1 - (exp_R . x)) / (1 + (exp_R . x)) ) )