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I am writing a document (mydoc.tex) that will include a section as a separate document (mysection.tex) using \input{mysection.tex}.

If I want to include something in the preamble (specifically \DeclareMathOperator{}{}), must this be done in the preamble of mydoc.tex or can I include it in mysection.tex?

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\DeclareMathOperator must go before \begin{document}, so there's no choice.

However, the concept of \input is quite easy:

\input{mysection}

will make TeX behave as if the entire contents of mysection.tex were at that point. It's best to put all personal commands in the document's preamble, so that you know precisely where to find them.

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