let Z be open Subset of REAL; ( Z c= dom (ln * arcsin) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
arcsin . x > 0 ) implies ( ln * arcsin is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arcsin) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) ) ) )
assume that
A1:
Z c= dom (ln * arcsin)
and
A2:
Z c= ].(- 1),1.[
and
A3:
for x being Real st x in Z holds
arcsin . x > 0
; ( ln * arcsin is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arcsin) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) ) )
A4:
for x being Real st x in Z holds
ln * arcsin is_differentiable_in x
then A5:
ln * arcsin is_differentiable_on Z
by A1, FDIFF_1:9;
for x being Real st x in Z holds
((ln * arcsin) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x))
proof
let x be
Real;
( x in Z implies ((ln * arcsin) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) )
assume A6:
x in Z
;
((ln * arcsin) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x))
then A7:
(
- 1
< x &
x < 1 )
by A2, XXREAL_1:4;
(
arcsin is_differentiable_in x &
arcsin . x > 0 )
by A2, A3, A6, FDIFF_1:9, SIN_COS6:83;
then diff (
(ln * arcsin),
x) =
(diff (arcsin,x)) / (arcsin . x)
by TAYLOR_1:20
.=
(1 / (sqrt (1 - (x ^2)))) / (arcsin . x)
by A7, SIN_COS6:83
.=
1
/ ((sqrt (1 - (x ^2))) * (arcsin . x))
by XCMPLX_1:78
;
hence
((ln * arcsin) `| Z) . x = 1
/ ((sqrt (1 - (x ^2))) * (arcsin . x))
by A5, A6, FDIFF_1:def 7;
verum
end;
hence
( ln * arcsin is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arcsin) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) ) )
by A1, A4, FDIFF_1:9; verum