let Z be open Subset of REAL; ( Z c= dom ((id Z) (#) arccos) & Z c= ].(- 1),1.[ implies ( (id Z) (#) arccos is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) arccos) `| Z) . x = (arccos . x) - (x / (sqrt (1 - (x ^2)))) ) ) )
assume that
A1:
Z c= dom ((id Z) (#) arccos)
and
A2:
Z c= ].(- 1),1.[
; ( (id Z) (#) arccos is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) arccos) `| Z) . x = (arccos . x) - (x / (sqrt (1 - (x ^2)))) ) )
A3:
for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0
by FUNCT_1:18;
Z c= (dom (id Z)) /\ (dom arccos)
by A1, VALUED_1:def 4;
then A4:
Z c= dom (id Z)
by XBOOLE_1:18;
then A5:
id Z is_differentiable_on Z
by A3, FDIFF_1:23;
A6:
arccos is_differentiable_on Z
by A2, FDIFF_1:26, SIN_COS6:106;
for x being Real st x in Z holds
(((id Z) (#) arccos) `| Z) . x = (arccos . x) - (x / (sqrt (1 - (x ^2))))
proof
let x be
Real;
( x in Z implies (((id Z) (#) arccos) `| Z) . x = (arccos . x) - (x / (sqrt (1 - (x ^2)))) )
assume A7:
x in Z
;
(((id Z) (#) arccos) `| Z) . x = (arccos . x) - (x / (sqrt (1 - (x ^2))))
then A8:
(
- 1
< x &
x < 1 )
by A2, XXREAL_1:4;
(((id Z) (#) arccos) `| Z) . x =
((arccos . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff (arccos,x)))
by A1, A5, A6, A7, FDIFF_1:21
.=
((arccos . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff (arccos,x)))
by A5, A7, FDIFF_1:def 7
.=
((arccos . x) * 1) + (((id Z) . x) * (diff (arccos,x)))
by A4, A3, A7, FDIFF_1:23
.=
((arccos . x) * 1) + (((id Z) . x) * (- (1 / (sqrt (1 - (x ^2))))))
by A8, SIN_COS6:106
.=
(arccos . x) + (x * (- (1 / (sqrt (1 - (x ^2))))))
by A7, FUNCT_1:18
.=
(arccos . x) - (x * (1 / (sqrt (1 - (x ^2)))))
.=
(arccos . x) - (x / (sqrt (1 - (x ^2))))
by XCMPLX_1:99
;
hence
(((id Z) (#) arccos) `| Z) . x = (arccos . x) - (x / (sqrt (1 - (x ^2))))
;
verum
end;
hence
( (id Z) (#) arccos is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) arccos) `| Z) . x = (arccos . x) - (x / (sqrt (1 - (x ^2)))) ) )
by A1, A5, A6, FDIFF_1:21; verum