"science of the composition and production of plays," 1795, from French dramaturgie, from Greek dramatourgia, from drama (genitive dramatos; see drama) + -ourgia "a working," from ergon "work, activity" (from PIE root *werg- "to do").
1510s, "a composition presenting in dialogue a course of human action, the description of a story converted into the action of a play," from Late Latin drama "play, drama," from Greek drama (genitive dramatos) "action, deed; play, spectacle," from drāo "to do, make, act, perform" (especially some great deed, whether good or bad), which is of uncertain etymology.
Meaning "theatrical literature generally, drama as art" is from 1660s. Extended sense of "sequence of events or actions leading up to a climax" is by 1714. Drama queen "person who habitually responds to situations in a melodramatic way" is attested by 1992.
It is the hypothetical source of/evidence for its existence is provided by: Greek ergon "work," orgia "religious performances;" Armenian gorc "work;" Avestan vareza "work, activity;" Gothic waurkjan, Old English wyrcan "to work," Old English weorc "deed, action, something done;" Old Norse yrka "work, take effect."
Others are reading
<a href="https://www.etymonline.com/word/dramaturgy">Etymology of dramaturgy by etymonline</a>
Harper, D. (n.d.). Etymology of dramaturgy. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/dramaturgy