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Vedas did exist before Vyasa. Vedas were classified by Vyasa.

Are Vedangas the part of Vedas? If not, who is the author of Vedangas?

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    "Vedas are compiled by Vyasa" - Incorrect terminology. Vedas were 'divided' by Vyasa. Vyas literally means 'divide/partition'. Compiled means he brought together different pieces. That's not what he did. He divided a huge portion into 4 manageable partitions, and each of his 4 disciples propagated one partition. The Vedas themselves were discovered during Tapas by Rishis. This site says Upanishads have reference to Vedangas, so they are also part of Vedas only. – mar Jul 28 '20 at 3:32
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Vedangas are not part of Vedas unlike Brahmanas, Aaranyakas and principal Upanishads which are usually considered to be part of Vedas. However Vedangas are believed to be essential for studying and understanding Vedas and that's why it's called limbs of Vedas.

Vedangas are six in number and each of six Vedangas consist several texts. There are not single texts associated with each of Vedas, but there are bunch of texts associated with particular Vedanga. So, we may not be able to say that a single Rishi or Acharya authored whole of any of six Vedangas, however there are certain Rishis/Acharyas which can be said to be chief propagators of Vedangas.

I recommend to read or go through principal texts of each of Vedangas to find these propagators of Vedangas and you can do some research on chronology of those text to find out the oldest text and author of those texts. I'm here listing the primary authors of each of Vedanga texts, based on PrasthanaBheda of Madhusudana Saraswati which I quoted in this answer for description of Vedangas.

S/N Vedanga Propagator(s)
1 Shiksha Panini
▪ Other authors of Shiksha texts associated with different Vedas
2 Vyakaran ▪ Munitraya (Three Munis) which are:
 1. Panini who wrote Ashtadhyayi
 2. Katyayana who wrote Vritti on Panini's Ashtadhyayi
 3. Patanjali who wrote Mahabhashya on Ashtadhyayi and Katyayana's Vritti
3 Nirukta Yaska
4 Chhanda Pingala composed Chhandasutras
5 Jyotisha ▪ Aaditya, Garga and others
6 Kalpa ▪ AshvalAyana, shAnkhAyana, etc. (Rigveda)
▪ BodhAyana, Apastamba, kAtyAyana, etc (Yajurveda)
▪ LATyAyana, drAhyAyaNa, etc. (Samaveda)

(Note): This broadly includes Srautasutras, Grihyasutras, Shulbasutras and Dharmasutras, composed by many Acharyas; see the list of texts

Note: Pratishakhyas text are sometimes considered to be part of Shiksha, however usually those texts contain knowledge of Vyakarana and Nirukta also. These texts are associated with different Shaksha of Vedas.

Also note that the concepts of these Vedangas were already there before these works are written and expounded, for example, when we say Panini is a propagator of Sanskrit Vyakarana, it doesn't mean Vyakarana was not there before Panini. Accroding to scholars, there had existed schools of grammer before Panini also and Panini mentioned them in his work.

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