let Z be open Subset of REAL; :: thesis: ( 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 ; :: 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_1:1;

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)))

((ln ^) `| Z) . x = - (1 / (x * ((ln . x) ^2))) ) ) by A2, A4, FDIFF_2:22; :: thesis: verum

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 ; :: 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_1:1;

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

hence
( ln ^ is_differentiable_on Z & ( for x being Real st x in Z holds
let x be Real; :: thesis: ( x in Z implies ((ln ^) `| Z) . x = - (1 / (x * ((ln . x) ^2))) )

assume A6: x in Z ; :: thesis: ((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))) ; :: thesis: verum

end;assume A6: x in Z ; :: thesis: ((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))) ; :: thesis: verum

((ln ^) `| Z) . x = - (1 / (x * ((ln . x) ^2))) ) ) by A2, A4, FDIFF_2:22; :: thesis: verum