theorem :: FDIFF_11:63

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