Is the mitzvah of veshinantam levanecha and the mitzvah of chinuch the same, or are they separate mitzvos? Are these mitzvos min ha-Torah? Also, who’s obligation and/or right is it, the father’s or the mother’s? In a circumstance of a divorce does anything change in regards to the obligation and rights to the above named mitzvos?
The mitzvah of teaching our children Torah, and even of teaching others Torah, is derived from the verse veshinantam levanecha. Although the verse speaks specifically about children, the Sages include disciples, for a person’s disciples are considered his children. This is a Torah mitzvah (see Rambam 11; Semag 12; Chinuch 419).
According to some authorities, this is also the source for the mitzvah of chinuch, for in teaching our children Torah, we also teach them the ways of keeping mitzvos (see Rishonim above and see, for instance, Moadim U-zemanim 3:237). Others mention different verses for the obligation of chinuch, such as the verse vehoda’atam levanecha (cited by Chafetz Chaim, Kiyum Ha-Torah 5:41), or from the verse in Bereishis (18:19) concerning Avraham Avinu (lema’an asher yetzaveh es zar’o acharav…, cited by Meshech Chochmah).
Ha’amek Davar (Devarim 11:1), however, and other authorities (see Rashba, Megillah 19b; Chayei Adam 66:1), write that the obligation of chinuch is rabbnic in nature, some stating that it is midivrei kabbalah (the more stringent form of rabbinic mitzvos).
Based on the mentioned sources, and on a statement of the Gemara (Nazir 29a), poskim write that not only the mitzvah of teaching Torah, but even the mitzvah of chinuch falls specifically on the father. Of course, the mother has a moral obligation to bring up her child on the path of Torah, but the actual Torah (or rabbinic) obligation is primarily the father’s. This would not change for a divorced couple.
See also introduction to Hakan Vehilchosav, who cites different opinions and sources for the obligation of chinuch.