Firstly I think that answer is amazing and I learned a lot from it even though I disagree with a lot of what they are saying. Math stack exchange is really hig

Firstly I think that answer is amazing and I learned a lot from it even though I disagree with a lot of what they are saying. Math stack exchange is really high-quality sometimes and I think this is a great example. For me the two notations are a lot like positional argument ordering vs named function arguments in a language like python which offers both. I personally prefer Legrange’s notation most of the time but think they both have their place. Legrange notation emphasises the ordering of the arguments and makes their naming going into the function irrelevant. This makes a lot of sense to me in a lot of situations where you have some pure abstract function with abstract arguments and makes it a lot easier to reason about certain things (like the example given or the chain rule which the author uses as an example where the common Leibniz formulation literally only makes sense if you do an explicit substitution in the function which is often not really explained). On the other hand Leibniz notation makes a lot of sense to me in contexts like physics where the names of the function arguments are really important. You aren’t just picking your favourites as the author claims. For a very simple example, if I say a = \frac{d^2s}{dt^2}, every physicist knows a bunch. I am calculating accelleration as the second derivative of displacement with respect to time. “s” isn’t just my favourite letter today. So “named arguments” make a lot of sense here. Secondly, Leibniz notation gives us the “derivative operator”[1], which is often very useful. Say I’m trying to find some derivative that involves a bunch of intermediate working out steps. It’s tremendously convenient to be able to say “<something> times d by dx of <some other expression>” and come back and differentiate “some other expression” in a later step. In Lagrange notation I would need to let that second expression be some named intermediate function so I have something to hang my tick off, which is definitely a lot more of a pain. [1] I’m not far enough along in my calculus journey to know whether I’m saying this right so forgive me if my terminology is incorrect - hopefully you get the intent.