let Z be open Subset of REAL ; :: thesis: ( Z c= dom (ln ^ ) & ( for x being Real st x in Z holds
x > 0 ) & ( 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 A1: ( Z c= dom (ln ^ ) & ( for x being Real st x in Z holds
x > 0 ) & ( for x being Real st x in Z holds
ln . x <> 0 ) ) ; :: thesis: ( 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_1:11;
then A2: Z c= dom ln by A1, XBOOLE_1:1;
then A3: ( ln is_differentiable_on Z & ( for x being Real st x in Z holds
(ln `| Z) . x = 1 / x ) ) by Th19;
then A4: ln ^ is_differentiable_on Z by A1, 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; :: thesis: ( x in Z implies ((ln ^ ) `| Z) . x = - (1 / (x * ((ln . x) ^2 ))) )
assume A5: x in Z ; :: thesis: ((ln ^ ) `| Z) . x = - (1 / (x * ((ln . x) ^2 )))
then A6: ( x > 0 & ln . x <> 0 ) by A1;
A7: ln is_differentiable_in x by A3, A5, FDIFF_1:16;
((ln ^ ) `| Z) . x = diff (ln ^ ),x by A4, A5, FDIFF_1:def 8
.= - ((diff ln ,x) / ((ln . x) ^2 )) by A6, A7, FDIFF_2:15
.= - (((ln `| Z) . x) / ((ln . x) ^2 )) by A3, A5, FDIFF_1:def 8
.= - ((1 / x) / ((ln . x) ^2 )) by A2, A5, Th19
.= - (1 / (x * ((ln . x) ^2 ))) by XCMPLX_1:79 ;
hence ((ln ^ ) `| Z) . x = - (1 / (x * ((ln . x) ^2 ))) ; :: thesis: 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 A1, A3, FDIFF_2:22; :: thesis: verum