theorem :: FDIFF_5:17

for Z being open Subset of REAL
for g being PartFunc of REAL,REAL st not 0 in Z & Z c= dom (g (#) (sin * ((id Z) ^))) & g = #Z 2 holds
( g (#) (sin * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
((g (#) (sin * ((id Z) ^))) `| Z) . x = ((2 * x) * (sin . (1 / x))) - (cos . (1 / x)) ) )