let Z be open Subset of REAL; ( Z c= dom (((id Z) ^) (#) arctan) & Z c= ].(- 1),1.[ implies ( - (((id Z) ^) (#) arctan) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (((id Z) ^) (#) arctan)) `| Z) . x = ((arctan . x) / (x ^2)) - (1 / (x * (1 + (x ^2)))) ) ) )
set f = id Z;
assume that
A1:
Z c= dom (((id Z) ^) (#) arctan)
and
A2:
Z c= ].(- 1),1.[
; ( - (((id Z) ^) (#) arctan) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (((id Z) ^) (#) arctan)) `| Z) . x = ((arctan . x) / (x ^2)) - (1 / (x * (1 + (x ^2)))) ) )
A3:
Z c= dom (- (((id Z) ^) (#) arctan))
by A1, VALUED_1:8;
A4:
for x being Real st x in Z holds
(id Z) . x = x
by FUNCT_1:18;
Z c= (dom ((id Z) ^)) /\ (dom arctan)
by A1, VALUED_1:def 4;
then A5:
Z c= dom ((id Z) ^)
by XBOOLE_1:18;
A6:
not 0 in Z
then A8:
( (id Z) ^ is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) ^) `| Z) . x = - (1 / (x ^2)) ) )
by FDIFF_5:4;
A9:
arctan is_differentiable_on Z
by A2, SIN_COS9:81;
A10:
((id Z) ^) (#) arctan is_differentiable_on Z
by A1, A2, A6, SIN_COS9:129;
then A11:
(- 1) (#) (((id Z) ^) (#) arctan) is_differentiable_on Z
by A3, FDIFF_1:20;
for x being Real st x in Z holds
((- (((id Z) ^) (#) arctan)) `| Z) . x = ((arctan . x) / (x ^2)) - (1 / (x * (1 + (x ^2))))
proof
let x be
Real;
( x in Z implies ((- (((id Z) ^) (#) arctan)) `| Z) . x = ((arctan . x) / (x ^2)) - (1 / (x * (1 + (x ^2)))) )
assume A12:
x in Z
;
((- (((id Z) ^) (#) arctan)) `| Z) . x = ((arctan . x) / (x ^2)) - (1 / (x * (1 + (x ^2))))
then A13:
((id Z) ^) (#) arctan is_differentiable_in x
by A10, FDIFF_1:9;
A14:
(id Z) ^ is_differentiable_in x
by A8, A12, FDIFF_1:9;
A15:
arctan is_differentiable_in x
by A9, A12, FDIFF_1:9;
((- (((id Z) ^) (#) arctan)) `| Z) . x =
diff (
(- (((id Z) ^) (#) arctan)),
x)
by A11, A12, FDIFF_1:def 7
.=
(- 1) * (diff ((((id Z) ^) (#) arctan),x))
by A13, FDIFF_1:15
.=
(- 1) * (((arctan . x) * (diff (((id Z) ^),x))) + ((((id Z) ^) . x) * (diff (arctan,x))))
by A14, A15, FDIFF_1:16
.=
(- 1) * (((arctan . x) * ((((id Z) ^) `| Z) . x)) + ((((id Z) ^) . x) * (diff (arctan,x))))
by A8, A12, FDIFF_1:def 7
.=
(- 1) * (((arctan . x) * (- (1 / (x ^2)))) + ((((id Z) ^) . x) * (diff (arctan,x))))
by A6, A12, FDIFF_5:4
.=
(- 1) * ((- ((arctan . x) * (1 / (x ^2)))) + ((((id Z) ^) . x) * ((arctan `| Z) . x)))
by A9, A12, FDIFF_1:def 7
.=
(- 1) * ((- (((arctan . x) * 1) / (x ^2))) + ((((id Z) ^) . x) * (1 / (1 + (x ^2)))))
by A2, A12, SIN_COS9:81
.=
(- 1) * ((- ((arctan . x) / (x ^2))) + ((((id Z) . x) ") * (1 / (1 + (x ^2)))))
by A5, A12, RFUNCT_1:def 2
.=
(- 1) * ((- ((arctan . x) / (x ^2))) + ((1 / x) * (1 / (1 + (x ^2)))))
by A4, A12
.=
(- 1) * ((- ((arctan . x) / (x ^2))) + ((1 * 1) / (x * (1 + (x ^2)))))
by XCMPLX_1:76
.=
((arctan . x) / (x ^2)) - (1 / (x * (1 + (x ^2))))
;
hence
((- (((id Z) ^) (#) arctan)) `| Z) . x = ((arctan . x) / (x ^2)) - (1 / (x * (1 + (x ^2))))
;
verum
end;
hence
( - (((id Z) ^) (#) arctan) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (((id Z) ^) (#) arctan)) `| Z) . x = ((arctan . x) / (x ^2)) - (1 / (x * (1 + (x ^2)))) ) )
by A3, A10, FDIFF_1:20; verum