theorem :: FDIFF_6:30

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