theorem :: FDIFF_4:37

for a, b being Real
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (sin * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( sin * f is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * f) `| Z) . x = a * (cos . ((a * x) + b)) ) )