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Pretty much every proof of the product or chain rules presented today revolve around the definition of the derivative as a limit (e.g. this post).

However, when Newton/Leibniz were developing calculus, they would not have had access to the concepts of limits. How, then, were the product and chain rules proved correct? Or was it just generally accepted that, if calculus worked, then the product and chain rules would just have to be in the form they were?

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    $\begingroup$ Did you look at en.wikipedia.org/wiki/Product_rule? It discusses Leibniz's approach. Any book on the history of calculus (there are a few) should provide some more details. $\endgroup$ – KCd Dec 31 '14 at 8:56
  • $\begingroup$ The chain rule in Leibniz notation is $da/dc=(da/db)(db/dc)$, which makes it look pretty obvious if you think of the d's as infinitesimals. You do have to worry about the distinction between the derivative and the ratio of the infinitesimals. Today we would describe this as taking the standard part of the ratio. Back then, they would have described this in terms of adequality or neglecting squares of infinitesimals. $\endgroup$ – Ben Crowell Jan 2 '15 at 14:54
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    $\begingroup$ related: mathoverflow.net/questions/181422/… $\endgroup$ – Ben Crowell Jan 2 '15 at 14:54
  • $\begingroup$ I believe G.H. Hardy was the first to express doubt to the classical proof of the chain rule (appearing when the inner function has infinitely many zeros). $\endgroup$ – Mr AD Aug 28 '15 at 17:40
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    $\begingroup$ The chain rule is not really even a rule when using differentials. An identity like $d(x^2) = 2xdx$ holds for any $x$, so that, for instance, $d\left(\left(\sin\theta\right)^2\right)=2\sin\theta d\left(\sin\theta\right)=2\sin\theta \cos\theta d\theta$. It would have been used implicitly. $\endgroup$ – user3339 Dec 6 '15 at 2:12
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This is not a complete answer, but the chain rule apparently was not even stated explicitly until 1797, by Lagrange. So says this reference by Rodríguez & Fernández. Footnote 5 in the paper reads:

As far as we can tell, the first “modern” version of the chain rule appears in Lagrange’s 1797 Théorie des fonctions analytiques, (Lagrange, J. L., 1797, §31, pp. 29); it also appears in Cauchy’s 1823 Résumé des Leçons données a L’École Royale Polytechnique sur Le Calcul Infinitesimal, (Cauchy, A. L., 1899, Troisième Leçon, pp. 25).

This footnote appears in section 2 of the paper, titled "History of the Chain Rule". According to this section, the chain rule is nowhere explicitly stated in Euler's books on analysis, nor even the notion of a composite function. (Wikipedia agrees with this, but their source seems to be the paper just mentioned.)

The chain rule appears implicitly in a memoir by Leibniz in 1676 (according to these authors, who cite The Early Mathematical Manuscripts of Leibniz, translated by J.M. Child). The idea seems to be the free use of differentials, presumably something like this computation: $$ d\sqrt{a+bz+cz^2}=\frac{b+2cz}{2\sqrt{a+bz+cz^2}}dz $$ Differentials are treated by Leibniz as infinitesimal differences. In L’Hospital's 1696 textbook Analyse des infiniment petits, the rule $dx^r=rx^{r-1}dx$ is given (our authors even use the word "proved", though they don't say how). L'Hospital then uses it pretty much the way a modern textbook would use the chain rule.

In short, the free use of Leibnizian differentials can serve the same purpose as the chain rule.

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