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Are the following words acceptable in formal writing : fashionista and chauffeur?

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"chauffeur" is OK to use at any level of English, though the word itself implies a grander or posher occupation than e.g. "driver". A rich or important person might have a "chauffeur".

"fashionista" has a slightly jargony or journalistic or frivolous feel. It may not be appropriate in serious formal writing, or academic writing, say. It would be OK in formal writing if you deliberately want to create a certain feel.