let Z be open Subset of REAL; ( Z c= dom (ln (#) arccot) & Z c= ].(- 1),1.[ implies ( ln (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) ) ) )
A1:
right_open_halfline 0 = { g where g is Real : 0 < g }
by XXREAL_1:230;
assume that
A2:
Z c= dom (ln (#) arccot)
and
A3:
Z c= ].(- 1),1.[
; ( ln (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) ) )
A4:
arccot is_differentiable_on Z
by A3, Th82;
Z c= (dom ln) /\ (dom arccot)
by A2, VALUED_1:def 4;
then A5:
Z c= dom ln
by XBOOLE_1:18;
A6:
for x being Real st x in Z holds
x > 0
then
for x being Real st x in Z holds
ln is_differentiable_in x
by TAYLOR_1:18;
then A7:
ln is_differentiable_on Z
by A5, FDIFF_1:9;
A8:
for x being Real st x in Z holds
diff (ln,x) = 1 / x
for x being Real st x in Z holds
((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2)))
proof
let x be
Real;
( x in Z implies ((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) )
assume A9:
x in Z
;
((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2)))
then ((ln (#) arccot) `| Z) . x =
((arccot . x) * (diff (ln,x))) + ((ln . x) * (diff (arccot,x)))
by A2, A7, A4, FDIFF_1:21
.=
((arccot . x) * (1 / x)) + ((ln . x) * (diff (arccot,x)))
by A8, A9
.=
((arccot . x) * (1 / x)) + ((ln . x) * ((arccot `| Z) . x))
by A4, A9, FDIFF_1:def 7
.=
((arccot . x) * (1 / x)) + ((ln . x) * (- (1 / (1 + (x ^2)))))
by A3, A9, Th82
.=
((arccot . x) / x) - ((ln . x) / (1 + (x ^2)))
;
hence
((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2)))
;
verum
end;
hence
( ln (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) ) )
by A2, A7, A4, FDIFF_1:21; verum