theorem :: FDIFF_9:29

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