let Z be open Subset of REAL ; :: thesis: ( 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
; :: thesis: ( 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 A7:
ln * arccos is_differentiable_on Z
by A1, FDIFF_1:16;
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;
:: thesis: ( x in Z implies ((ln * arccos ) `| Z) . x = - (1 / ((sqrt (1 - (x ^2 ))) * (arccos . x))) )
assume A8:
x in Z
;
:: thesis: ((ln * arccos ) `| Z) . x = - (1 / ((sqrt (1 - (x ^2 ))) * (arccos . x)))
then A9:
(
- 1
< x &
x < 1 )
by A2, XXREAL_1:4;
A10:
arccos is_differentiable_in x
by A2, A8, FDIFF_1:16, SIN_COS6:108;
arccos . x > 0
by A3, A8;
then diff (ln * arccos ),
x =
(diff arccos ,x) / (arccos . x)
by A10, TAYLOR_1:20
.=
(- (1 / (sqrt (1 - (x ^2 ))))) / (arccos . x)
by A9, SIN_COS6:108
.=
- ((1 / (sqrt (1 - (x ^2 )))) / (arccos . x))
by XCMPLX_1:188
.=
- (1 / ((sqrt (1 - (x ^2 ))) * (arccos . x)))
by XCMPLX_1:79
;
hence
((ln * arccos ) `| Z) . x = - (1 / ((sqrt (1 - (x ^2 ))) * (arccos . x)))
by A7, A8, FDIFF_1:def 8;
:: thesis: 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_1:16; :: thesis: verum