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Re: "Transparency" with respect to differentiation

> And, additionaly, how could I define a standard form of this object, 
> that returns a 'nicer' version (however maybe a bit less unambiguous), 

I found out the answer to the last question of the OP (i.e. myself):

Advanced Topic "Low-Level Input and Output Rules", 2.9.17, gives an
example how to change the output format, still using the mathematical
definition. If I want to display the function as prefix "d":

quat /:
     StandardForm] :=
       StandardForm], MakeBoxes[arg]}]


d asdf

Letting this "quat" or "d" let pass the differentiation to its argument
is still an open question.

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