3. Vyākaraṇa - Vedanga

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Patañjali's Mahābhāṣya

Patañjali's 2nd-century BCE Mahābhāṣya is another important ancient text in Vyākaraṇa scholarship. It is not a full commentary on everything Pāṇini wrote in Aṣṭādhyāyī, but it is more a commentary on Kātyāyana's text on grammar called Varttikas, as well as the ideas of Vyadi. While Kātyāyana's additions have survived, Vyadi have not.

The Kātyāyana's text reflects an admiration for Pāṇini, an analysis of his rules, their simplification and refinement. The differences between the grammar rules of Pāṇini and of Kātyāyana may be because of historical changes to Sanskrit language over the centuries, state Howard Coward and K Kunjunni Raja.

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Bhartṛhari's Vākyapadīya

The Vākyapadīya of Bhartṛhari is a treatise on the philosophy of language, building on the insights of prior Vyākaraṇa scholarship.

According to Bhartṛhari, states Scharfstein, all thought and all knowledge are "words", every word has an outward expression and inward meaning. A word may have a definition in isolation but it has meaning only in the context of a sentence. Grammar is a basic science in the Hindu traditions, explains Scharfstein, where it is externally expressed as relations between words, but ultimately internally understood as reflecting relations between the different levels of reality. Word is considered a form of energy in this Hindu text, one with the potential to transform a latent mind and realize the soul. Language evolves to express the transient material world first, and thereon to express feelings, the human desire for meaning in life and the spiritual inner world.

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Roots of words

In Yāska's time, nirukta "etymology" was in fact a school which gave information of formation of words, the etymological derivation of words. According to the nairuktas or "etymologists", all nouns are derived from a verbal root. Yāska defends this view and attributes it to Śākaṭāyana. While others believed that there are some words which are "Rudhi Words". 'Rudhi" means custom. 
Meaning they are a part of language due to custom, and a correspondence between the word and the thing if it be a noun or correspondence between an act and the word if it be a verb root. Such word can not be derived from verbal roots. Yāska also reports the view of Gārgya, who opposed Śākaṭāyana who held that certain nominal stems were 'atomic' and not to be derived from verbal roots

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Influence

This section needs expansion. You can help by adding to it. (March 2016)
The Vyākaraṇa texts have been highly influential on Hindu philosophies. The concept of a sentence (vakya) defined by Pāṇini, for instance, influenced and was similar to Jaimini, the later era founder of Mīmāṃsā school of Hindu philosophy. However, ritual-focussed Mimamsa school scholars were generally opposed to central ideas of the Hindu Grammarians, while others Hindu schools such as Vedanta championed them.

Pāṇini's work on Vyākaraṇa has been called by George Cardona as "one of the greatest monuments of human intelligence".

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Notes

In addition to explicit references to Vyākaraṇa, Rigveda has numerous embedded Riddle hymns, a few of which ancient and medieval Hindu scholars interpreted to be referring to linguistics and grammar. 
For example, the riddle verse 4.58.3 of the Rigveda states,
"Four horns, three feet, two heads and seven hands he has.
The bull is thrice bound and roars.
Great is the god who has entered the man". – Rigveda 4.58.3

Patañjali interprets this riddle as follows, state Annette Wilke and Oliver Moebus: The four horns represent the four parts of speech: nouns, verbs, prefixes and particles. The three feet are the three main tenses: present, future, past; the two heads are the conventional and the etymological meaning of a word; the seven hands are the seven cases in Sanskrit; the three places where the roaring bull is bound are the three resonating spaces - the chest, the neck and the head' and the great god in riddle is the word.

The earliest secondary literature on the primary text of Pāṇini are by Kātyāyana (~3rd century BCE) and Patañjali (~2nd century BCE).
Bhartṛhari is now dated to have lived no later than the 5th century CE, but there is a mention in a Chinese text by I-tsing that Bhartrihari died in 651/652 CE.