what is the difference between a duke and a grand duke?

Question

is a duchy a territory inside the grand duchy. and is a grand duchy a territory inside the entire country

Answer 1

There is only one Grand Duke in he world who reigns over a Grand Duchy - Luxembourg - and the title is unique.
He his classed as "royalty" as opposed to many Dukes who are just nobility.

Answer 2

A duke is a nobleman, historically of highest rank below the king, and usually controlling a duchy. The title comes from the Latin dux, which had the sense of "military commander" and was employed both by the Germanic peoples themselves and by the Roman authors covering them to refer to their war leaders.

In the Middle Ages, the title signified first, among the Germanic monarchies. The dukes were the rulers of the provinces and the superiors of the counts in the cities, and later, in the feudal monarchies, the highest-ranking peers of the king. There were, however, variants of these meanings, and there were even sovereign princes employing ducal titles.

In the Modern Age, it has become a nominal rank without an actual principality. It is still the highest titular peerage in France, Portugal, Spain, the United Kingdom, and Italy.

A woman who holds in her own right the title to such duchy or dukedom, or is the wife of a duke, is normally styled duchess. However, Queen Elizabeth II is known as the Duke of Lancaster and the Duke of Normandy in the Channel Islands, and is called that on their coinage.

The title Grand Duke (in Latin, Magnus Dux; in Spanish, Gran Duque; in Russian, Великий князь; in German, Großherzog, Italian Gran Duca; in French, Grand-duc; in Portuguese, Grão-duque; in Finnish, Suurherttua; in Polish, Wielki Książę; in Hungarian, nagyherceg;in Swedish, Storhertig; in Dutch, Groothertog; in Danish, Storhertug, in Lithuanian, Didysis Kunigaikštis) used in Western Europe and particularly in Germanic countries for provincial sovereigns, is of a protocolary rank below King but higher than a sovereign Duke (Herzog) or Fürst (usually but confusingly translated as "Prince").

The feminine form is Grand Duchess. A Grand Duke's principality (state or merely titular) is called a Grand Duchy.

Grand Duke is the usual and established translation of Grand Prince in languages which do not have separate words meaning prince for (1) children of a monarch, and (2) monarch (sovereign or like) princes. English and French use Grand Duke also in this way.

The title Grand Duke as translation of Grand Prince and the proper title Grand Duke have clearly different meanings and a separate background. Compare with the article Grand Prince.