'bio-derived' originates from Greek and Latin/English combining elements: specifically the Greek word 'bios' (meaning 'life'), used as the combining form 'bio-', combined with the English past participle 'derived' (from Latin 'derivare').

'bio-' as a combining form comes from Greek 'bios' and entered scientific English via New Latin and Modern usage; 'derived' comes from Latin 'derivare' (via Old French and Middle English 'derive'); the compound 'bio-derived' is a modern scientific/technical formation in English.

Initially the parts meant 'life' ('bio-') and 'to draw off or obtain from a source' ('derive'); over time the compound settled into the specific modern sense 'obtained from biological sources (rather than fossil or synthetic sources)'.