theorem :: FDIFF_11:69

for n being Element of NAT
for Z being open Subset of REAL st Z c= dom ((1 / n) (#) ((#Z n) * (arctan ^))) & Z c= ].(- 1),1.[ & n > 0 & ( for x being Real st x in Z holds
arctan . x <> 0 ) holds
( (1 / n) (#) ((#Z n) * (arctan ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2)))) ) )