Where did Leibniz explore the product rule of differential calculus?

Question

In what book/letter did Gottfried Wilhelm Leibniz explore the product rule as part of differential calculus?

Answer 1

It is discussed in multiple manuscripts, letters and publications from 1675 to 1701.

According to Fracois Ziegler's post on MO Did Leibniz really get the Leibniz rule wrong?, Leibniz originally thought $d(uv)=du\,dv$ in a special case, but corrected his mistake the same month in the manuscript Methodi tangentium inversae exempla (November 11, 1675). Later the same month the correct general rule $d\overline xy = d\overline{xy}-xd\overline y$ appears in Pro methodo tangentium inversa et aliis tetragonisticis specimina et inventa (November 27, 1675), where he calls it "a really noteworthy theorem and a general one for all curves".

It reappears in Elementa calculi novi... (1680), which was a draft of the Acta Eruditorum paper Nova Methodus pro Maximis et Minimis... (October 1684) linked in Victor Blasjo's answer, his first publication on the new calculus. Even there he does not prove the product rule, or other propositions, but justifications appears e.g. in a letter to Wallis from March 30, 1699, based on what we now call the "method of exhaustion", and in Cum prodiisset atque increbuisset Analysis mea infinitesimalis (1701), based on Leibniz's signature "continuity principle":"In any supposed continuous transition, ending in any terminus, it is permissible to institute a general reasoning, in which the final terminus may also be included". Both methods are used in Justification du Calcul des infinitesimales (1701).

Leibniz's evolution on infinitesimals, and the product rule in particular, is described in On the Attempts made by Leibniz to Justify his Calculus by Horvath:

"From about 1680, the leibnizian calculus step by step becomes more developed, and more consolidated, and the variable feature of the infinitely small quantity comes into the foreground. From that time on we may observe in Leibniz's works the more conscious attempts to outline the concept, and the use of the infinitely small quantities. For example, in his manuscript Elementa, the quantities $_1D_2C,\,_2C_3D, ...$ are conceived as "incrementa momentanea" of the line segment BC8. In the same manuscript he states that $d\overline{xy}$, that is $d(xy)$, is the difference between two proximate terms of the finite, two-dimensional, geometrical variable $xy$. One of these terms is the variable xy, and the other is $(x+dx)(y+dy)$, so that $d(xy)$ is equal to $(x+dx)(y+dy) - xy$.

[...] The above-mentioned manuscript Elementa is a preliminary draft of Leibniz's famous article entitled Nova Methodus pro maximis et minimis (see [4]). It is worth observing that there is a sharp distinction between the two conceptions of infinitely small quantities used by Leibniz in these two papers. In the draft, (see [3]), as we have seen, the line segments $DC$ were thought by Leibniz to be infinitely small, that is "incrementa momentanea". In the final version [4], however, Leibniz does not use differentials but only differences in the sense of fixed, small, finite quantities. Leibniz supposedly does not use the term "infinitely small" in his article [4] in order to avoid controversies which most likely would have arisen in connection with this notion

[...] It is known that Leibniz did not prove the propositions which occurred in his first publication of his calculus (see [4]). Now let's turn to the discussion how Leibniz justifies, for example, the rule of differentiation for a product $xy$ using incomparable quantities, that is alluding to the method of Archimedes. The quantity $d(xy)$ is equal to the quantity $(x+dx) (y+dy) - xy$. (So $d(xy)$ is the same as the difference between two adjacent $xy$, of which let one be $xy$, the other $(x+dx)(y+dy)$.) During calculations the quantity $dxdy$ can be omitted since this term is incomparably smaller with respect to the others, thus we obtain the sound result $d(xy) = xdy + ydx$. That reasoning can be treated in archimedian style, stating that the error will be smaller than any given positive quantity. Leibniz points out ([30], p. 63) that he does not discuss the question of whether or not inassignable quantities are fictitious concepts or not, because it is sufficient that these quantities serve for the shortening of the reasoning.".

Horvath also describes the continuity principle justification, but it is more involved, so I will not reproduce it here. Leibniz's manuscipts on calculus are collected in Historia et origo calculi differentialis a G. G. Leibnitio conscripta (1846) edited by Gerhardt, see also Archive's version.

Answer 2

Leibniz states the product rule in his first paper on the calculus (1684). It's in the middle of the fist page (page 467) as can be seen here: https://www.maa.org/press/periodicals/convergence/mathematical-treasure-leibnizs-papers-on-calculus-differential-calculus and also in English translation (top of page 2) here: http://www.17centurymaths.com/contents/Leibniz/nova1.pdf