'exude' originates from Latin, specifically the word 'exsūdāre' (or 'exsudare'), where 'ex-' meant 'out' and 'sūdāre' meant 'to sweat' or 'to ooze'.

'exude' changed from the Latin verb 'exsūdāre' (or Medieval Latin 'exsudare') into Middle English (via borrowing and regular sound changes) and eventually became the modern English word 'exude'.

Initially, it meant 'to sweat or to ooze out' (a literal physical discharge); over time it also developed the figurative sense 'to display or emit (a quality) strongly', which is common in modern usage.