theorem :: FDIFF_4:56

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