In a sincere, respectful, and deeply religious manner; with genuine dedication and reverence.
From 'devout' (adjective, from Old French 'devot' and Latin 'devotus') plus '-ly' adverb suffix. Standard formation that's been in English since the 14th century.
Unlike most '-ly' adverbs that merely describe manner, 'devoutly' carries moral weight—calling someone 'devoutly religious' isn't just describing how they practice faith, it's affirming their sincerity, showing how adverbs can encode judgment.
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