I retracted my previous answer in the comments since it was actually for the product rule, not the chain rule. Sorry about that. After performing another search, I was able to find some links that might be helpful:
When/How were the product and chain rules first proved?
See Top Answer by Michael Weiss.
The link that Weiss provides is no longer active, but I believe I found a working link:
Rodriguez, Omar Hernandez and Lopez Fernandez, Jorge M. (2010) “A semiotic reflection on the didactics of the Chain rule,” The Mathematics Enthusiast: Vol. 7 : No. 2, Art. 10.
Starting in Section 2: “History of the Chain Rule,” Rodriguez, states
It may come as a surprise to the reader that nowhere in Analyse des
infiniment petits, (L’Hospital, 1696), is the chain rule stated
It may be even more surprising to realize that the
statement of the chain rule is so absent in all of Euler’s analysis
books, Introductio in analysin infinitorum, (Euler, 1748, Vol. 1),
(Euler, 1748, Vol. 2), and Institutiones calculi differentialis,
(Euler, 1755). Furthermore, Euler did define the notion of a function
in (Euler, L., 1748, Vol. 1), but he never treated the topic of the
composition of functions in any of his writings, (Euler, 1748, Vol.
1), (Euler, 1748, Vol. 2) and (Euler, 1755).
As far as we can tell,
the first mention of the Chain Rule in the literature of calculus
seems to be due to Leibniz (Child, 2007, p. 126), and it appears in a
1676 memoir (with various mistakes) in which he calculated...
[Footnote] 5 - 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,
Here's the citation for the reference text:
Child, J. M. (translator), Leibniz, G.W. (Author). (2007). The Early Mathematical Manuscripts of Leibniz. Merchant Books.
So, to answer your title question, according to Child, the manuscript is simply titled,
"Manuscript, dated Nov., 1676" with the section titled "Calculus Tangentium differentialis [Differential calculus of tangents]." However, as Rodriguez and Fernandez point out in their paper, don't expect to see in Leibniz's writings the chain rule theorem in the form that we now know and recognize.
Edit: Regarding the "publication date," from what I've gathered from a cursory read of Child, it appears that Leibniz didn't publish these particular manuscripts himself, so I would be hesitant to put an emphasis on such dates. Instead, the concern is when Leibniz put these ideas pen-to-paper in his notes. From what I've gathered, it seems that Carl Immanuel Gerhardt is the one who compiled Leibniz's notes and published them in a series of texts dated 1846, 1848, and 1855, but I wouldn't give these dates much weight.