Haven't seen anywhere your derivation. here are two different Sanskrit scholars interpretations. Surendranath Dasgupta writes in his History of Indian Philosophy (Volume 1) - available here on print page 38, digital page 32 - http://consciouslivingfoundation.org/ebooks/13/CLF-HistoryOfIndianPhilosophy.pdf
The meaning of the word Upanisad.
The word Upanisad is derived from the root sad with the prefix ni (to sit), and Max Muller says that the word originally meant the act of sitting down near a teacher and of submissively listening to him. In his introduction to the Upanisads he says, "The history and the genius of the Sanskrit language leave little doubt that Upanisad meant originally session, particularly a session consisting of pupils, assembled at a respectful distance round their teacher [Footnote ref 1]." Deussen points out that the word means "secret" or
"secret instruction," and this is borne out by many of the passages of the Upanisads themselves. Max Muller also agrees that the word was used in this sense in the Upanisads [Footnote ref 2]. There we find that great
injunctions of secrecy are to be observed for the communication of the doctrines, and it is said that it should only be given to a student or pupil who by his supreme moral restraint and noble desires proves himself deserving to hear them. Sankara however, the great Indian exponent of the Upanisads, derives the word from the root sad to destroy and supposes that it is so called because it destroys inborn ignorance and leads to salvation by revealing the right knowledge. But if we compare the many texts in which the word Upanisad
occurs in the Upanisads themselves it seems that Deussen's meaning is fully justified [Footnote ref 3].
Chandradhar Sharma writes in his A Critical Survey of Indian Philosophy, p 5 on pdf (available here - https://archive.org/details/IndianPhilosophyACriticalSurvey):
We now come to the Upanisads which are the concluding portion as
well as the cream of the Veda and are therefore rightly called ‘Vedanta’.
The word ‘Upanisad’ is derived from the root ‘sad’ which means (i) to
sit down, (ii) to destroy and (iii) to loosen. ‘Upa’ means ‘near by’ and ‘ni’
means ‘devotedly*. The word therefore means the sitting down of the
disciple near his teacher in a devoted manner to receive instruction about
the highest Reality which loosens all doubts and destroys all ignorance
of the disciple. Gradually the word came to signify any secret teaching
about Reality and it is used by the Upanisads in this sense (rahasya or