[nimonian]

OC's point is that the chain rule for partial derivatives shouldn't be assumed because the ordinary chain rule holds, there's more depth to it than that, and the proof is harder than you might instinctively expect based on the ordinary chain rule.

It's epistemically acceptable to understand these both as "the chain rule" once we're satisfied they've both been proved, and apply liberal amounts of synecdoche from there (and I don't think OC disagrees with you on that).

[rd11235]

Actually by 'ordinary chain rule' I am referring to what you're referring to as 'the chain rule for partial derivatives'. It seems like backprop follows very quickly even from that, but it does not.