theorem :: FDIFF_4:55

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