I have seen many (even well-known) people in Buddhist discourse say that “dependent origination is an extension of/equivalent to the 2nd Noble Truth”. And sometimes that the 3rd Noble Truth mirrors the second part - dependent cessation.
I was not able to find any explicit sutta support for this (can you?), and I think it is a modern extrapolation that comes from wanting to “categorize” information “under” the 4 Noble Truths, and maybe from a desire to help rationalize things for beginners. I don’t think this categorization is too useful.
However, if you wanted to somehow map it to one Noble Truth, then I think the 1st Noble Truth is better anyway, with only a 1/12th sized hint of the Second and Third.
Here’s a side by side of dependent origination and the Noble Truths:
1 avijjā | 3rd? 4th?
… | 1st?
7 vedanā | 2nd? “nandi” (except non-pleasurable feelings)
8 taṇhā | 2nd and 3rd
9 upādāna | 1st: “(>in summary<) pañcupādānakkhandhā dukkhā”
10 bhava
11 jāti | 1st: “jāti pi dukkhā” (“birth is also suffering”)
12 jarā | 1st: “jarā pi dukkhā”
12 maraṇaṁ | 1st: “maraṇaṁ pi dukkhaṁ”
12 sokaparidevadukkhadomanassupāyāsā | 1st: “soka… pi dukkhā”
Arguably, 4. nāmarūpa, 5. saḷāyatana, 6. phassa, 7. vedanā are mentioned in the 1st with “piya” and “icchati” - your loved ones, what you hold dear, what you want, is suffering
At least 3 of the links or explicitly 5 terms are mentioned in the First Noble Truth, compared to 1 link and 1 term for the Second Noble Truth. But, does the First NT capture the essence of dependent origination more than the second (and third)? That’s hard to say since taṇhā certainly matters, but so does avijjā, and upādāna, and jāti…, they all play a role. With the ending of vedanā, there’s the ending of taṇhā, so taṇhā isn’t a “bigger” factor in D.O. as it totally dependent on vedanā. Taṇhā takes up just 1/12th, but maybe it’s one of the more “noticeable” factors, easier to see why it’s connected with the delusions of selfishness and permanence, so it’s spotlighted in the 4 Noble Truths. Maybe the spirit of the second is “finding a cause”, similar to “dependent” origination, but in that case, I’d say avijjā is primary, especially in terms of a dedicated practice, since it is the first cause, the root, and it’s what one can directly change with mental development - with wisdom, from concentration and stillness, from virtue.
You could also see this in terms of specific conditioned realizations. Like that each Noble Truth and dependent origination are specific realizations, totaling to 5. Perhaps you would have to know what exactly is suffering (1st NT; last links) in order to know what causes it (2nd; 8th link) (or vice versa), therefore those two Noble Truths would not exclude each other in the process of realizing dependent origination.
In the end, I don’t think dependent origination should be seen as classified under or an expansion of any particular Noble Truth, but that it’s just one explanation by the Buddha of the truth. Some of his explanations end up being deeply related between each other, but it only goes as far as complex relationships in this case, just fused together, not categorized under or as. These doctrines are also fused with the ones on kamma and the ones on rebirth, for example.
P.S. just to be a little more controversial, “Nth Noble Truth” isn’t the best name scheme. They were almost always called, respectively: suffering noble truth, suffering-origin noble truth, suffering-cessation noble truth, and suffering-cessation-going-path noble truth. I found 74 results for “dukkhasamudayaṁ ariyasaccaṁ” and 2 for “dutiyaṁ ariyasaccaṁ”.
Do you disagree? Opinions welcome for me. With mettā for you all.