theorem :: FDIFF_4:53

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