theorem :: FDIFF_4:57

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