Is saying "אתה חננתנו"/הבדלה on מוצ''ש is it מן התורה or not?
On one hand I remember hearing it is מן התורה as it part of "זכור" but on the other hand I've heard its only מדרבנן
Please provide sources.
As with many questions, we can say that it is a machlokes. The Rambam says that it is from the Torah while Rabbeinu Tam says that it is rabbinic.
. Is havdalah a biblical or rabbinic obligation?
Opinion of the Rambam. The Rambam (Hilchot Shabbat 29:1) writes that the biblical commandment of “zachor et yom hashabbat l’kadsho” consists of mentioning the praises of Shabbat when it begins (with kiddush) and when it ends (with havdalah), clearly indicating that the obligation to say havdalah is biblical in nature.
Opinion of Rabeinu Tam. The Rabeinu Tam is cited in the response of the Rosh (11:3) as having ruled that the obligation to recite havdalah is only rabbinic in nature.
Even according to the Rambam, the particular method used is actually mid'rabbanan
The source for havdalah b’tfilah. The gemara (Berachot 33a) records that originally the rabbis instituted the recitation of havdalah in davening. Subsequently, when Jews became more affluent and were able to afford wine, the rabbis instead required that it be recited over a cup of wine. In later times, when the financial standing of the Jewish community slipped, the rabbis resorted to the earlier decree to recite havdalah during davening. In order to guarantee that the method with which one is supposed to recite havdalah not change constantly with the affluence of the community, the rabbis made a permanent decree to recite havdalah during davening, and insisted that one who says havdalah in davening still recite havdalah over a cup of wine. (See Rashba ad loc. and Shulchan Aruch Harav 294:2)
Yeshiva Har Etzion has the following
Does the obligation of havdala originate from the Torah, or was it instituted by Chazal? Some Rishonim indicate that havdala constitutes a rabbinic obligation, whereas others imply that it originates from the Torah. If, indeed, havdala is required according to Torah law, the question arises as to the Biblical source of this obligation. The Rambam writes in Hilkhot Shabbat (29:1):
"There is a mitzvat asei from the Torah to declare the sanctity of the Shabbat day, as it says, 'Remember the day of Shabbat, to make it holy,' meaning, speak about it in terms of praise and sanctity. One must speak of it when it enters and when it departs: when it enters – through the kiddush of the day; when it departs – through havdala."
This passage implies that the havdala obligation stems from the mitzva of "zakhor" ("Remember the Shabbat day… " – Shemot 20:8). The Maggid Mishneh notes that other Rishonim derive the havdala obligation from the verse in Parashat Shemini, "To distinguish between the sacred and the mundane" (Vayikra 10:10).
This issue regarding the Biblical source of the mitzva may yield several important ramifications. For example, the Minchat Chinukh (mitzva 31) raises the question of why we make no mention of the Exodus in havdala as we do in kiddush. If the obligation originates from the verse in Parashat Shemini, the answer is easily resolved: since the requirement of havdala stems from a different source from kiddush, we should not expect its laws to parallel those of kiddush. If, however, we view "zakhor" – the source of the obligation of kiddush – as the source of havdala, as well, then these two should indeed follow the same format. The Minchat Chinukh leaves this question unresolved. Rav Tzvi Pesach Frank (Har Tzvi, O.C. 157) suggests that we perhaps fulfill the obligation to mention the Exodus through the recitation of arvit on Motza'ei Shabbat before havdala. Rav Shlomo Zalman Auerbach (cited in Shemirat Shabbat Ke-hilkhata, chapter 58, note 18) claimed that even if we view havdala as part of the mitzva of zakhor, this would not necessarily require mentioning the Exodus in havdala. It suffices to mention the Exodus while reciting kiddush, and there is no need to mention it again in havdala.