theorem :: FDIFF_7:19

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