English 'duke' comes through Old French 'duc' from Latin 'dux' (leader), derived from 'dūcere' (to lead), from PIE *dewk- (to lead, to pull).
A nobleman of the highest hereditary rank below a prince; also, the ruler of a duchy or small state.
From Proto-Indo-European *deuk- ("to lead"), via Latin dux (genitive ducis, "leader, commander"), from the verb ducere ("to lead, to guide"). The PIE root *deuk- also yields Latin ducere and its many derivatives. The word entered Old French as duc and was carried into Middle English after the Norman Conquest (c. 1066). Originally a military rank — dux was the Roman title for a general commanding a regional force
The Italian word 'doge' — the title of the rulers of Venice and Genoa — comes from the same Latin 'dux.' The Doge's Palace in Venice is literally the 'Leader's Palace.' And the medieval gold coin called a 'ducat' takes its name from the Latin 'ducatus' (duchy), because it was originally minted by the Duchy of Apulia.