let Z be open Subset of REAL; :: thesis: ( Z c= dom (sin * arccot) & Z c= ].(- 1),1.[ implies ( sin * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (sin * arccot) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( sin * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) ) )
A3: for x being Real st x in Z holds
sin * arccot is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies sin * arccot is_differentiable_in x )
assume A4: x in Z ; :: thesis: sin * arccot is_differentiable_in x
A5: sin is_differentiable_in arccot . x by SIN_COS:64;
arccot is_differentiable_on Z by A2, SIN_COS9:82;
then arccot is_differentiable_in x by A4, FDIFF_1:9;
hence sin * arccot is_differentiable_in x by A5, FDIFF_2:13; :: thesis: verum

end;

then A6: sin * arccot is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2)))

proof

let x be Real; :: thesis: ( x in Z implies ((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) )
assume A7: x in Z ; :: thesis: ((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2)))
A8: arccot is_differentiable_on Z by A2, SIN_COS9:82;
then A9: arccot is_differentiable_in x by A7, FDIFF_1:9;
A10: sin is_differentiable_in arccot . x by SIN_COS:64;
((sin * arccot) `| Z) . x = diff ((sin * arccot),x) by A6, A7, FDIFF_1:def 7
.= (diff (sin,(arccot . x))) * (diff (arccot,x)) by A9, A10, FDIFF_2:13
.= (cos . (arccot . x)) * (diff (arccot,x)) by SIN_COS:64
.= (cos . (arccot . x)) * ((arccot `| Z) . x) by A7, A8, FDIFF_1:def 7
.= (cos . (arccot . x)) * (- (1 / (1 + (x ^2)))) by A2, A7, SIN_COS9:82
.= - ((cos . (arccot . x)) / (1 + (x ^2))) ;
hence ((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) ; :: thesis: verum

end;

hence ( sin * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) ) ) by A1, A3, FDIFF_1:9; :: thesis: verum