theorem Th11: :: FDIFF_7:11

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