theorem Th31: :: FDIFF_6:31

for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((#Z 2) * (f - exp_R)) & ( for x being Real st x in Z holds
f . x = 1 ) holds
( (#Z 2) * (f - exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z 2) * (f - exp_R)) `| Z) . x = - ((2 * (exp_R . x)) * (1 - (exp_R . x))) ) )