theorem Th13: :: FDIFF_4:13

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 (ln * (f1 + (c (#) f2))) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a + (b * x) & (f1 + (c (#) f2)) . x > 0 ) ) holds
( ln * (f1 + (c (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 + (c (#) f2))) `| Z) . x = (b + ((2 * c) * x)) / ((a + (b * x)) + (c * (x |^ 2))) ) )