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