let Z be open Subset of REAL; ( Z c= dom (ln ^) & ( for x being Real st x in Z holds
ln . x <> 0 ) implies ( ln ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((ln ^) `| Z) . x = - (1 / (x * ((ln . x) ^2))) ) ) )
set f = ln ;
assume that
A1:
Z c= dom (ln ^)
and
A2:
for x being Real st x in Z holds
ln . x <> 0
; ( ln ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((ln ^) `| Z) . x = - (1 / (x * ((ln . x) ^2))) ) )
dom (ln ^) c= dom ln
by RFUNCT_1:1;
then A3:
Z c= dom ln
by A1;
then A4:
ln is_differentiable_on Z
by Th19;
then A5:
ln ^ is_differentiable_on Z
by A2, FDIFF_2:22;
for x being Real st x in Z holds
((ln ^) `| Z) . x = - (1 / (x * ((ln . x) ^2)))
proof
let x be
Real;
( x in Z implies ((ln ^) `| Z) . x = - (1 / (x * ((ln . x) ^2))) )
assume A6:
x in Z
;
((ln ^) `| Z) . x = - (1 / (x * ((ln . x) ^2)))
then A7:
(
ln . x <> 0 &
ln is_differentiable_in x )
by A2, A4, FDIFF_1:9;
((ln ^) `| Z) . x =
diff (
(ln ^),
x)
by A5, A6, FDIFF_1:def 7
.=
- ((diff (ln,x)) / ((ln . x) ^2))
by A7, FDIFF_2:15
.=
- (((ln `| Z) . x) / ((ln . x) ^2))
by A4, A6, FDIFF_1:def 7
.=
- ((1 / x) / ((ln . x) ^2))
by A3, A6, Th19
.=
- (1 / (x * ((ln . x) ^2)))
by XCMPLX_1:78
;
hence
((ln ^) `| Z) . x = - (1 / (x * ((ln . x) ^2)))
;
verum
end;
hence
( ln ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((ln ^) `| Z) . x = - (1 / (x * ((ln . x) ^2))) ) )
by A2, A4, FDIFF_2:22; verum