How were Ad-Darooriyyaat Al-Khams derived? Does it have any basis in the Quran and Hadith, or is it just a scholarly opinion?

All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.

The five necessities (Ad-Dharooriyyaat Al-Khams) are recognized and protected not only by the Islamic law (sharee'ah), but also by all the (divine) laws. The fact that the Islamic sharee'ah recognizes and protects them is one of the indisputably established facts of Islam that stands in no need of proof or deduction. It is not based on a specific piece of evidence; rather, it is derived through the Istiqraa' (induction) of  a group of the mutawaatir texts of sharee'ah (i.e. texts that are reported by a group of narrators from a group of narrators, generation after generation and so forth, and they are all trustworthy people and it is absolutely impossible for them to agree on a lie) that indicate it.

Ash-Shaatibi wrote about the five fundamental necessities and their supportive evidence, "The Muslim nation - or rather all nations - has collectively agreed on the fact that the sharee'ah aims at the preservation of the five necessities, which are: religion, life, mind, offspring and property. Knowledge of this is an essential part of religion. This was not established through a specific textual evidence; rather, it was known to conform to the sharee'ah through numerous pieces of evidence that cannot possibly be listed in one chapter." [Al-Muwaafaqaat]

Hence, the recognition of the five necessities by the sharee'ah is not a matter of Ijtihaad (scholarly reasoning). Rather, it is a decisive, well-established fact, and it is agreed upon by the consensus of the scholars.

Allaah knows best.