theorem :: FDIFF_7:5

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