theorem :: FDIFF_7:12

for Z being open Subset of REAL st Z c= dom ((1 / 2) (#) ((#Z 2) * arcsin)) & Z c= ].(- 1),1.[ holds
( (1 / 2) (#) ((#Z 2) * arcsin) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arcsin)) `| Z) . x = (arcsin . x) / (sqrt (1 - (x ^2))) ) )