theorem :: FDIFF_9:28

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