This is a basic formula in mechanics, which determines the acceleration of a particle performing uniform circular motion.
By who first derived it? In Newton's Principa, what one can find is that
$$a \propto v^2/r $$
History of Science and Mathematics Stack Exchange is a question and answer site for people interested in the history and origins of science and mathematics. It only takes a minute to sign up.Sign up to join this community
(a) In answer, first a comment about proportionalities and equalities. Many 17th-century writers, including Huygens and Newton, commonly used proportions rather than equalities, especially to avoid doing what would now be calculation of ratios between physical quantities of unlike kind, such ratios being considered somehow illegitimate. This old tradition and theory of proportion dating back to Euclid is explained by N Guicciardini in "Mathematics and the New Sciences", chapter 8, (esp. sec. 8.3.2 (p.232-233) in 'Oxford Handbook of the History of Physics' ed J Buchwald et al., 2013).
(b) There is a chronology that seems to be widely accepted for the discoveries and publications of the $ v^2/r $ relation between speed, curvature and bending force or acceleration: and it is usually concluded that Newton and Huygens made their findings independently. Here is the chronology along with references to documents:
(1) Huygens' manuscript of 'De vi centrifuga' carries a date of 1659, and includes the $ v^2/r $ relation, stated verbally and in terms of proportion and centrifugal force, with its derivation: however, according to the 20th-c. editors of Huygens' Oeuvres complètes, Vol. XVI, at p.254, this work was not published until the appearance of the 'Opuscula posthuma' of 1703, with a re-impression in 1728.
(2) There is a Newton manuscript of about 1669 or before which derives the $ v^2/r $ relation in terms of proportions and of 'conatus' which can amount to centrifugal force -- rather than in terms of (modern) force or acceleration, concepts which were only clarified later -- and in the same ms Newton then employed the result to derive (from the Kepler three-halves power relation between orbital radius and period) an inverse-square law of centrifugal force for the approximated case of assumedly circular planetary orbits. The ms was not published by Newton, it was first published and discussed in modern literature by Rupert Hall in “Newton on the Calculation of Central Force,” Annals of Science 13 (1957), 62–71. The document was also reproduced, translated and commented in the Correspondence of Isaac Newton, vol.1 (1661-1675) (Cambridge, 1959), at pp.297-303 and plates facing pages 297,299. It also appeared in John Herivel's 'The Background to Newton's Principia' (Oxford, 1965), where pp.192-198 give commentary, Newton's diagram, text and an English translation from Latin.
(3) Huygens' book 'Horologium oscillatorium' (1673) (available in Latin at https://books.google.com/books?id=YgY8AAAAMAAJ&pg=PA159) contains in part V, p.159 onwards, a number of 'theorems' out of 'De vi centrifuga' which Huygens set out as conclusions but without derivation/proof. The result relevant to the present question is stated verbally and in terms of proportion and centrifugal force. It can be gathered from p.160, theorems II, III and IV. To show the style, what Huygens wrote is transcribed and translated here as:
(Original Latin): "II: Si duo mobilia aequalia, aequali celeritate ferantur, in circumferentiis inaequalibus; erunt eorum vires centrifugae in ratione contraria diametrorum.
III: Si duo mobilia aequalia in circumferentiis aequalibus ferantur, celeritate inaequali, sed utraque motu aequabili, quarum in his omnibus intelligi volumus; erit vis centrifuga velocioris, ad vim tardioris, in ratione duplicata celeritatum.
IV: Si mobilia duo aequalia, in circumferentiis inaequalibus circumlata, vim centrifugam aequalem habuerint; erit tempus circuitus in majori circumferentia, ad tempus circuitus in minori, in subdupla ratione diametrorum."
Translated: "II: If two equal movable objects are carried with equal speed around unequal circumferences, [then] their centrifugal forces will be in the inverse ratio of the diameters.
III: If two equal movable objects are carried around equal circumferences with unequal speeds, but each with equable motion, which we wish to be understood in all these theorems, [then] the centrifugal force of the faster, to the force of the slower, will be in the duplicate ratio of the speeds.
IV: If two equal movable bodies, carried around unequal circumferences, have equal centrifugal force, [then] the time of circuit around the greater circumference, to the time of circuit around the lesser, will be in the subduplicate ratio of the diameters."
(4) Newton's manuscript 'De Motu Corporum in gyrum' (1684) (more details are at (https://en.wikipedia.org/wiki/De_motu_corporum_in_gyrum)) derived the result as its Theorem 2, showing that for any given time-segment, the centripetal force (directed towards the center of a circle, treated as a center of attraction) is proportional to the square of the arc-length traversed, and inversely proportional to the radius. This short precursor of the 'Principia' was made available to members of the Royal Society by its entry as a copy in late 1684/early 1685 into the Royal Society's register books.
(6) The 1703 posthumous edition of Huygens' works eventually printed 'De vi centrifuga' with his derivation of the relationships quoted above (at p.409 onwards).