theorem :: FDIFF_9:11

for a, b, c being Real
for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom (cosec * (f1 + (c (#) f2))) & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = a + (b * x) ) holds
( cosec * (f1 + (c (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * (f1 + (c (#) f2))) `| Z) . x = - (((b + ((2 * c) * x)) * (cos . ((a + (b * x)) + (c * (x ^2))))) / ((sin . ((a + (b * x)) + (c * (x ^2)))) ^2)) ) )