[Math] Did Leibniz really get the Leibniz rule wrong

Question

A couple of posts ([1], [2]) on matheducators.SE seem to suggest that Leibniz originally got the wrong form for the product rule, perhaps thinking that $(fg)'=f'g'$. Is there any actual historical evidence for this?

It seems particularly hard to believe that he would have made the hypothesis $(fg)'=f'g'$. We would then have $x'=(1x)'=(1')(x')=0$. And presumably anyone inventing calculus would take $(x^2)'$ to be a prototypical problem, and would realize pretty early on that $(x^2)'\ne (x')(x')=1$. It's also pretty trivial to disprove this conjecture based on dimensional analysis or scaling.

There is some discussion on this Wikipedia talk page, with some sources cited, but it appears to be inconclusive.

Answer 1

In the manuscript "Determinationum progressio in infinitum" (pp. 668-675 of Sämtliche Schriften und Briefe, Reihe VII, Band 3, Teil C, available in pdf here), Leibniz writes on p. 673 (with "$\sqcap$" in place of "$=$"):

$$ \odot = \overline{dt}\int\frac{a^2}{a^2 + t^2}. \quad\text{Hence}\quad \overline{d\odot} = \frac{a^2}{a^2 + t^2}\overline{d\overline{dt}} $$

This amounts to asserting that $d[uv] = dv\,du$ where $u=dt$ and $v=\int\frac{a^2}{a^2+t^2}$; and thus differentiating the product wrong, as the editors comment in footnote 14. On p. 668 they take this as grounds to date the manuscript early November 1675, since by November 11 he was pointing out this error (in "Methodi tangentium inversae exempla", quoted by Edwards in KConrad's comment above).

Addendum: The first time Leibniz gets his general rule right appears to be in "Pro methodo tangentium inversa et aliis tetragonisticis specimina et inventa" (dated 27 November 1675; pp. 361-371 of the same Sämtliche Schriften, Reihe VII, Band 5, Teil B; English translation here), where he writes on p. 365:

Therefore $d\overline xy = d\overline{xy}-xd\overline y$. Now this is a really noteworthy theorem and a general one for all curves.