let Z be open Subset of REAL; :: thesis: ( Z c= dom (ln * arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
arccot . x > 0 ) implies ( ln * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x))) ) ) )
assume that
A1: Z c= dom (ln * arccot) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
arccot . x > 0 ; :: thesis: ( ln * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x))) ) )
A4: for x being Real st x in Z holds
ln * arccot is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies ln * arccot is_differentiable_in x )
assume A5: x in Z ; :: thesis: ln * arccot is_differentiable_in x
arccot is_differentiable_on Z by A2, Th82;
then A6: arccot is_differentiable_in x by A5, FDIFF_1:9;
arccot . x > 0 by A3, A5;
hence ln * arccot is_differentiable_in x by A6, TAYLOR_1:20; :: thesis: verum

end;

then A7: ln * arccot is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x)))

proof

let x be Real; :: thesis: ( x in Z implies ((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x))) )
assume A8: x in Z ; :: thesis: ((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x)))
then A9: - 1 < x by A2, XXREAL_1:4;
arccot is_differentiable_on Z by A2, Th82;
then A10: arccot is_differentiable_in x by A8, FDIFF_1:9;
A11: x < 1 by A2, A8, XXREAL_1:4;
arccot . x > 0 by A3, A8;
then diff ((ln * arccot),x) = (diff (arccot,x)) / (arccot . x) by A10, TAYLOR_1:20
.= (- (1 / (1 + (x ^2)))) / (arccot . x) by A9, A11, Th76
.= - ((1 / (1 + (x ^2))) / (arccot . x))
.= - (1 / ((1 + (x ^2)) * (arccot . x))) by XCMPLX_1:78 ;
hence ((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x))) by A7, A8, FDIFF_1:def 7; :: thesis: verum

end;

hence ( ln * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x))) ) ) by A1, A4, FDIFF_1:9; :: thesis: verum