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The concept itself, along with the rest of the vowel gradation processes, has its origin in an early stage of the Proto-Indo-European language.
V?ddhi in Sanskrit
The general phenomenon of vowel gradation, including v?dhhi formation has been extensively studied and documented as part of Sanskrit's vigorous grammatical tradition, most importantly in the Ahadhyay? of the grammarian Pini.
In modern Indo-European linguistics it is used in Pini's sense, but not restricted to Sanskrit but applicable to the Indo-European languages in general as well as to the Proto-Indo-European (PIE) language from which this feature was inherited:
*b?r?-[f] (zero grade of the reconstructed verb meaning "to carry")
A v?ddhi-derivation or v?ddhi-derivative is a word that is derived by such lengthening, a type of formation very common in Sanskrit, but also attested in other languages. Such derivatives signify "of, belonging to, descended from". An example:
PIE *swé?uro- "father-in-law" (Vedic Sanskrit?vá?ura-) -> *sw??uró- "relating to one's father-in-law" (Vedic ?v??ura- "relating to one's father-in-law", Old High Germansw?gur "brother-in-law")
Derivatives that are formed by inserting a full grade (as opposed to a lengthened grade) vowel into the "wrong" position of a zero grade are also called v?ddhi-derivations:
PIE *diw-, zero grade of *dy?u-s "sky"-> *deyw-os "god, sky god" (Vedic devás, Latindeus, etc.)
^in Sanskrit, a -tí-nomen actionis formed from the verbal root v?dh-/vardh- 'to grow'