theorem :: FDIFF_4:50

for Z being open Subset of REAL st Z c= dom (sin + ((1 / 2) (#) ((#Z 2) * sin))) & ( for x being Real st x in Z holds
( sin . x > 0 & sin . x < 1 ) ) holds
( sin + ((1 / 2) (#) ((#Z 2) * sin)) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin + ((1 / 2) (#) ((#Z 2) * sin))) `| Z) . x = ((cos . x) |^ 3) / (1 - (sin . x)) ) )