let Z be open Subset of REAL; ( Z c= dom (cot * arctan) & Z c= ].(- 1),1.[ implies ( cot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) ) )
assume that
A1:
Z c= dom (cot * arctan)
and
A2:
Z c= ].(- 1),1.[
; ( cot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) )
A3:
for x being Real st x in Z holds
cot * arctan is_differentiable_in x
then A6:
cot * arctan is_differentiable_on Z
by A1, FDIFF_1:9;
for x being Real st x in Z holds
((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2))))
proof
let x be
Real;
( x in Z implies ((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) )
assume A7:
x in Z
;
((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2))))
then
arctan . x in dom cot
by A1, FUNCT_1:11;
then A8:
sin . (arctan . x) <> 0
by FDIFF_8:2;
then A9:
cot is_differentiable_in arctan . x
by FDIFF_7:47;
A10:
arctan is_differentiable_on Z
by A2, SIN_COS9:81;
then A11:
arctan is_differentiable_in x
by A7, FDIFF_1:9;
((cot * arctan) `| Z) . x =
diff (
(cot * arctan),
x)
by A6, A7, FDIFF_1:def 7
.=
(diff (cot,(arctan . x))) * (diff (arctan,x))
by A11, A9, FDIFF_2:13
.=
(- (1 / ((sin . (arctan . x)) ^2))) * (diff (arctan,x))
by A8, FDIFF_7:47
.=
- ((1 / ((sin . (arctan . x)) ^2)) * (diff (arctan,x)))
.=
- ((1 / ((sin . (arctan . x)) ^2)) * ((arctan `| Z) . x))
by A7, A10, FDIFF_1:def 7
.=
- ((1 / ((sin . (arctan . x)) ^2)) * (1 / (1 + (x ^2))))
by A2, A7, SIN_COS9:81
.=
- (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2))))
by XCMPLX_1:102
;
hence
((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2))))
;
verum
end;
hence
( cot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) )
by A1, A3, FDIFF_1:9; verum