theorem :: FDIFF_6:24

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