Theorems first published in textbooks?

Long ago, I proved that every derivation from $C^{k+1}$ functions to $C^k$ functions is given by a $C^k$ vector field. (The same fact with $\infty$ in place of $k$ and $k+1$ is, of course, classical.) The first (and only, as far as I know) publication is in the book "Manifolds, Tensor Analysis, and Applications" by Abraham, Marsden, and Ratiu (p.235).

The story behind this is that, at the time, I was sharing an office with Bill Floyd; Tudor Ratiu, whose office was just down the hall, was working on this book. Of course, he knew about the $C^\infty$ version of the result, but he stopped by to ask Bill about the $C^k$ version, and I happened to be there too. Neither Bill nor I knew anything about it, but later (the same evening, I think, but my memory may be playing tricks here) I worked out a proof. When I told Tudor about it, he asked if he could put it into the book, and I said sure. I think the proof in the book is more streamlined than my original argument.

I recall that, and Wikipedia independently confirms that L'Hôpital's rule first appeared in a textbook, apparently the first textbook on differential calculus: Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes published by Guillaume de l'Hôpital and made up of content mostly provided by Johann Bernoulli, who was on retainer to l'Hôpital, more or less, for this purpose.

It happened to me once. While visiting the Institute for Advanced Studies at the Hebrew University of Jerusalem in 1976-77 I answered a question from the preliminary manuscript of volume 1 of Lindenstrauss-Tzafriri by constructing Banach spaces not isomorphic to Hilbert spaces all of whose subspaces have the approximation property. They replaced the question with one of my examples in the published book. I delayed writing the paper, which appeared several years later (1980). Actually I wrote the paper only because L-T had included the simplest rather than the most interesting example (which had the property that every subspace of every quotient has a Schauder basis).