let a, b be Real; for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (f (#) arcsin) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( f (#) arcsin is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arcsin) `| Z) . x = (a * (arcsin . x)) + (((a * x) + b) / (sqrt (1 - (x ^2)))) ) )
let Z be open Subset of REAL; for f being PartFunc of REAL,REAL st Z c= dom (f (#) arcsin) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( f (#) arcsin is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arcsin) `| Z) . x = (a * (arcsin . x)) + (((a * x) + b) / (sqrt (1 - (x ^2)))) ) )
let f be PartFunc of REAL,REAL; ( Z c= dom (f (#) arcsin) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (a * x) + b ) implies ( f (#) arcsin is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arcsin) `| Z) . x = (a * (arcsin . x)) + (((a * x) + b) / (sqrt (1 - (x ^2)))) ) ) )
assume that
A1:
Z c= dom (f (#) arcsin)
and
A2:
Z c= ].(- 1),1.[
and
A3:
for x being Real st x in Z holds
f . x = (a * x) + b
; ( f (#) arcsin is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arcsin) `| Z) . x = (a * (arcsin . x)) + (((a * x) + b) / (sqrt (1 - (x ^2)))) ) )
Z c= (dom f) /\ (dom arcsin)
by A1, VALUED_1:def 4;
then A4:
Z c= dom f
by XBOOLE_1:18;
then A5:
f is_differentiable_on Z
by A3, FDIFF_1:23;
A6:
arcsin is_differentiable_on Z
by A2, FDIFF_1:26, SIN_COS6:83;
for x being Real st x in Z holds
((f (#) arcsin) `| Z) . x = (a * (arcsin . x)) + (((a * x) + b) / (sqrt (1 - (x ^2))))
proof
let x be
Real;
( x in Z implies ((f (#) arcsin) `| Z) . x = (a * (arcsin . x)) + (((a * x) + b) / (sqrt (1 - (x ^2)))) )
assume A7:
x in Z
;
((f (#) arcsin) `| Z) . x = (a * (arcsin . x)) + (((a * x) + b) / (sqrt (1 - (x ^2))))
then A8:
(
- 1
< x &
x < 1 )
by A2, XXREAL_1:4;
((f (#) arcsin) `| Z) . x =
((arcsin . x) * (diff (f,x))) + ((f . x) * (diff (arcsin,x)))
by A1, A5, A6, A7, FDIFF_1:21
.=
((arcsin . x) * ((f `| Z) . x)) + ((f . x) * (diff (arcsin,x)))
by A5, A7, FDIFF_1:def 7
.=
((arcsin . x) * a) + ((f . x) * (diff (arcsin,x)))
by A3, A4, A7, FDIFF_1:23
.=
((arcsin . x) * a) + ((f . x) * (1 / (sqrt (1 - (x ^2)))))
by A8, SIN_COS6:83
.=
(a * (arcsin . x)) + (((a * x) + b) * (1 / (sqrt (1 - (x ^2)))))
by A3, A7
.=
(a * (arcsin . x)) + (((a * x) + b) / (sqrt (1 - (x ^2))))
by XCMPLX_1:99
;
hence
((f (#) arcsin) `| Z) . x = (a * (arcsin . x)) + (((a * x) + b) / (sqrt (1 - (x ^2))))
;
verum
end;
hence
( f (#) arcsin is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arcsin) `| Z) . x = (a * (arcsin . x)) + (((a * x) + b) / (sqrt (1 - (x ^2)))) ) )
by A1, A5, A6, FDIFF_1:21; verum