>From searching the net (1), I gather that 
log(x).conjugate(x).diff(x)

should yield
(log(x)/x).conjugate()

but Sage cannot evaluate such differentiated conjugates of functions:
sage: ex=log(x).conjugate()
sage: ex=log(x).conjugate(); ex
conjugate(log(x))
sage: ex.diff(x)
D[0](conjugate)(log(x))/x
sage: ex.diff(x).subs(x=-1/2)
-2*D[0](conjugate)(I*pi + log(1/2))
sage: ex.diff(x).subs(x=-1/2).n()
---------------------------------------------------------------------------
TypeError: cannot evaluate symbolic expression numerically

which should just be
sage: conjugate(log(x)/x).subs(x=-1/2).n()
1.38629436111989 + 6.28318530717959*I
if I understand it correctly. Am I wrong?

(1) https://answers.yahoo.com/question/index?qid=20120210002819AAtlqub

PS: Note also that Wolfram gives 
d/dx(log(x)^conjugate) = (Conjugate'(log(x)))/x

when asked for the derivative of conjugate of log(x).


Regards,

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