theorem :: FDIFF_6:41

for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (ln * f)) & f = f1 + (2 (#) sin) & ( for x being Real st x in Z holds
( f1 . x = 1 & f . x > 0 ) ) holds
( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . x))) ) )