let Z be open Subset of REAL; ( Z c= dom (ln * arccos) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
arccos . x > 0 ) implies ( ln * arccos is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arccos) `| Z) . x = - (1 / ((sqrt (1 - (x ^2))) * (arccos . x))) ) ) )
assume that
A1:
Z c= dom (ln * arccos)
and
A2:
Z c= ].(- 1),1.[
and
A3:
for x being Real st x in Z holds
arccos . x > 0
; ( ln * arccos is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arccos) `| Z) . x = - (1 / ((sqrt (1 - (x ^2))) * (arccos . x))) ) )
A4:
for x being Real st x in Z holds
ln * arccos is_differentiable_in x
then A5:
ln * arccos is_differentiable_on Z
by A1, FDIFF_1:9;
for x being Real st x in Z holds
((ln * arccos) `| Z) . x = - (1 / ((sqrt (1 - (x ^2))) * (arccos . x)))
proof
let x be
Real;
( x in Z implies ((ln * arccos) `| Z) . x = - (1 / ((sqrt (1 - (x ^2))) * (arccos . x))) )
assume A6:
x in Z
;
((ln * arccos) `| Z) . x = - (1 / ((sqrt (1 - (x ^2))) * (arccos . x)))
then A7:
(
- 1
< x &
x < 1 )
by A2, XXREAL_1:4;
(
arccos is_differentiable_in x &
arccos . x > 0 )
by A2, A3, A6, FDIFF_1:9, SIN_COS6:106;
then diff (
(ln * arccos),
x) =
(diff (arccos,x)) / (arccos . x)
by TAYLOR_1:20
.=
(- (1 / (sqrt (1 - (x ^2))))) / (arccos . x)
by A7, SIN_COS6:106
.=
- ((1 / (sqrt (1 - (x ^2)))) / (arccos . x))
by XCMPLX_1:187
.=
- (1 / ((sqrt (1 - (x ^2))) * (arccos . x)))
by XCMPLX_1:78
;
hence
((ln * arccos) `| Z) . x = - (1 / ((sqrt (1 - (x ^2))) * (arccos . x)))
by A5, A6, FDIFF_1:def 7;
verum
end;
hence
( ln * arccos is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arccos) `| Z) . x = - (1 / ((sqrt (1 - (x ^2))) * (arccos . x))) ) )
by A1, A4, FDIFF_1:9; verum