Apart from not murdering and not eating meat from a live animal, where are the Noachide laws derived from? Those aforementioned are mentioned in Genesis 9:6, but where (outside the laws of Moses) is it a given to follow the other laws?
The logic is given by Rambam in Mishneh Torah Melakhim uMilchamot 9:1:
על ששה דברים נצטווה אדם הראשון [...] הוסיף לנח אבר מן החי שנאמר אך בשר בנפשו דמו לא תאכלו נמצאו שבע מצות
Six precepts were commanded to Adam. [...] The prohibition against eating flesh from a living animal was added for Noah, as it is stated (Bereishit 9:4): 'Nevertheless, you may not eat flesh with its life, which is its blood.' Thus there are seven mitzvot.
So you have to go back to Adam to see these six laws that are listed with their sources in Bereishit Rabbah 16:6. Five are based on the method called gezeirah shavah that derives teachings based on common words in different verses. Bereishit 2:16 was divided into six parts by the sages referring to six commandments:
- idolatry – וַיְצַו: based on the common word צו also found in Hoshea 5:11
- blasphemy – 'ה: Hashem alludes to Vayikra 24:16
- judges – אֱלֹקִים: the other divine name, Elokim alludes to Shemot 22:27
- (bloodshed – עַל־הָאָדָם: man alludes to Bereishit 9:6)
- incest – לֵאמֹר: saying alludes to Yirmiyahu 3:1
- theft – מִכֹּל עֵץ־הַגָּן אָכֹל תֹּאכֵל: Of every tree of the garden thou mayest freely eat alludes to the fact that one needs a permission from an owner to take her/his property.
(See also Sanhedrin 56b)
Another great article from MyJewishLearning.com regarding the question.