let Z be open Subset of REAL ; :: thesis: ( Z c= ].(- 1),1.[ implies ( arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(arccot `| Z) . x = - (1 / (1 + (x ^2 ))) ) ) )
assume A1: Z c= ].(- 1),1.[ ; :: thesis: ( arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(arccot `| Z) . x = - (1 / (1 + (x ^2 ))) ) )
then A2: arccot is_differentiable_on Z by Th72, FDIFF_1:34;
for x being Real st x in Z holds
(arccot `| Z) . x = - (1 / (1 + (x ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies (arccot `| Z) . x = - (1 / (1 + (x ^2 ))) )
assume A4: x in Z ; :: thesis: (arccot `| Z) . x = - (1 / (1 + (x ^2 )))
then A5: ( - 1 <= x & x <= 1 ) by A1, XXREAL_1:4;
thus (arccot `| Z) . x = diff arccot ,x by A2, A4, FDIFF_1:def 8
.= - (1 / (1 + (x ^2 ))) by A5, Th74 ; :: thesis: verum

end;

hence ( arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(arccot `| Z) . x = - (1 / (1 + (x ^2 ))) ) ) by A1, Th72, FDIFF_1:34; :: thesis: verum