In a derogatory manner; with words or tone that show disrespect, contempt, or disapproval.
From 'derogatory' + '-ily' adverbial suffix (where '-y' becomes '-ily'). This creates the adverbial form of the more common 'derogatory' adjective.
This is the most natural-sounding way to express the idea—'he spoke derogatorily about her background' flows better than the alternatives. It's one of those adverbs that sounds harsh precisely because of its formal construction.
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