let n be Element of NAT ; :: thesis:
let Z be open Subset of REAL; :: thesis:
assume A1: ( Z c= dom ((#Z n) * arccot) & Z c= ].(- 1),1.[ ) ; :: thesis:
then A2: Z c= dom (- ((#Z n) * arccot)) by VALUED_1:8;
A3: (#Z n) * arccot is_differentiable_on Z by A1, SIN_COS9:92;
then A4: (- 1) (#) ((#Z n) * arccot) is_differentiable_on Z by A2, FDIFF_1:20;
for x being Real st x in Z holds
((- ((#Z n) * arccot)) `| Z) . x = (n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))

proof

let x be Real; :: thesis:
assume A5: x in Z ; :: thesis:
then A6: ( - 1 < x & x < 1 ) by A1, XXREAL_1:4;
arccot is_differentiable_on Z by A1, SIN_COS9:82;
then A7: arccot is_differentiable_in x by A5, FDIFF_1:9;
A8: (#Z n) * arccot is_differentiable_in x by A3, A5, FDIFF_1:9;
((- ((#Z n) * arccot)) `| Z) . x = diff ((- ((#Z n) * arccot)),x) by A4, A5, FDIFF_1:def 7
.= (- 1) * (diff (((#Z n) * arccot),x)) by A8, FDIFF_1:15
.= (- 1) * ((n * ((arccot . x) #Z (n - 1))) * (diff (arccot,x))) by A7, TAYLOR_1:3
.= (- 1) * ((n * ((arccot . x) #Z (n - 1))) * (- (1 / (1 + (x ^2))))) by A6, SIN_COS9:76
.= (n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2)) ;
hence ((- ((#Z n) * arccot)) `| Z) . x = (n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2)) ; :: thesis:

end;

hence ( - ((#Z n) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((- ((#Z n) * arccot)) `| Z) . x = (n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2)) ) ) by A2, A3, FDIFF_1:20; :: thesis: