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

proof

let x be Real; :: thesis: ( x in Z implies cos * arctan is_differentiable_in x )
assume A4: x in Z ; :: thesis: cos * arctan is_differentiable_in x
A5: cos is_differentiable_in arctan . x by SIN_COS:63;
arctan is_differentiable_on Z by A2, SIN_COS9:81;
then arctan is_differentiable_in x by A4, FDIFF_1:9;
hence cos * arctan is_differentiable_in x by A5, FDIFF_2:13; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies ((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2))) )
assume A7: x in Z ; :: thesis: ((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2)))
A8: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A9: arctan is_differentiable_in x by A7, FDIFF_1:9;
A10: cos is_differentiable_in arctan . x by SIN_COS:63;
((cos * arctan) `| Z) . x = diff ((cos * arctan),x) by A6, A7, FDIFF_1:def 7
.= (diff (cos,(arctan . x))) * (diff (arctan,x)) by A9, A10, FDIFF_2:13
.= (- (sin . (arctan . x))) * (diff (arctan,x)) by SIN_COS:63
.= - ((sin . (arctan . x)) * (diff (arctan,x)))
.= - ((sin . (arctan . x)) * ((arctan `| Z) . x)) by A7, A8, FDIFF_1:def 7
.= - ((sin . (arctan . x)) * (1 / (1 + (x ^2)))) by A2, A7, SIN_COS9:81
.= - ((sin . (arctan . x)) / (1 + (x ^2))) ;
hence ((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2))) ; :: thesis: verum

end;

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