theorem :: FDIFF_11:65

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