let Z be open Subset of REAL ; :: thesis: ( Z c= dom (cos (#) arccot ) & Z c= ].(- 1),1.[ implies ( cos (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) arccot ) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2 ))) ) ) )
assume that
A1: Z c= dom (cos (#) arccot ) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( cos (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) arccot ) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2 ))) ) )
Z c= (dom cos ) /\ (dom arccot ) by A1, VALUED_1:def 4;
then A3: Z c= dom cos by XBOOLE_1:18;
for x being Real st x in Z holds
cos is_differentiable_in x by SIN_COS:68;
then A4: cos is_differentiable_on Z by A3, FDIFF_1:16;
A5: arccot is_differentiable_on Z by A2, SIN_COS9:82;
for x being Real st x in Z holds
((cos (#) arccot ) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies ((cos (#) arccot ) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2 ))) )
assume A6: x in Z ; :: thesis: ((cos (#) arccot ) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2 )))
((cos (#) arccot ) `| Z) . x = ((arccot . x) * (diff cos ,x)) + ((cos . x) * (diff arccot ,x)) by A1, A4, A5, A6, FDIFF_1:29
.= ((arccot . x) * (- (sin . x))) + ((cos . x) * (diff arccot ,x)) by SIN_COS:68
.= (- ((sin . x) * (arccot . x))) + ((cos . x) * ((arccot `| Z) . x)) by A5, A6, FDIFF_1:def 8
.= (- ((sin . x) * (arccot . x))) + ((cos . x) * (- (1 / (1 + (x ^2 ))))) by A2, A6, SIN_COS9:82
.= (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2 ))) ;
hence ((cos (#) arccot ) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2 ))) ; :: thesis: verum

end;

hence ( cos (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) arccot ) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2 ))) ) ) by A1, A4, A5, FDIFF_1:29; :: thesis: verum