theorem Th1: :: FDIFF_4:1

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