What is the proofs according to the Quran of using ahadith?
For instance, salah is mentioned but may not be derives as the salah we have today if not using ahadith.

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One proof is Sura Hashr verse 7. It states whatever the Prophet (s) commands you to do then take it and do it, and whatever he prohibits you from doing then avoid it. This basically means you must follow his teachings. This means you must study hadith to see what he commanded and prohibited so you can follow it. Also, all the verses that command us to obey the Prophet (s) also command us to know hadith so we can obey the Prophet (s).