theorem :: FDIFF_4:32

for a, b being Real
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((2 / (3 * b)) (#) ((#R (3 / 2)) * f)) & ( for x being Real st x in Z holds
( f . x = a + (b * x) & b <> 0 & f . x > 0 ) ) holds
( (2 / (3 * b)) (#) ((#R (3 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / (3 * b)) (#) ((#R (3 / 2)) * f)) `| Z) . x = (a + (b * x)) #R (1 / 2) ) )