The key clarification is in one of the comments: if you want to treat partial derivatives like fractions, you need to carry the "constant with respect to foo" modifier along with both nominator and denominator.

Once you do that, it's clear that you can't cancel "dx at constant z" with "dx at constant y" etc. And then the remaining logic works out nicely (see thermodynamics for a perfect application of this).

I’ve never liked to conflation with fractions. Abuse of notation. And it causes so much confusion.

Also integrals with “integrate f(x) dx” where people treat “dx” as some number than can be manipulated, when it’s more just part of the notation “integrate_over_x f(x)”

Sigh. These are sadly some kind of right-of-passage, or mathematical hazing. Sad.

I still don’t understand what "at constant something" means. I mean formally, mathematically, in a way where I don’t have to kinda guess what the result may be and rely on my poor intuitions and shoot myself continually in the foot in the process.

Does someone has a good explanation ?

i'm looking forward to the day calculus gets rewritten using more intuitive notations.

Everytime i manipulate dx i feel like walking on a minefield.

Am I missing something, I don't see how the examples are more "intuitive" as they just provide an allied example of using this?

My pain was always Hamiltonians and Legendre equations for systems because the lecturer believed in learn by rote rather than explaining something that I'm sure for him was simply intuitive.

Why would you even tell in the first places derivatives are simply fractions ? They’re not, unless in some very specific physical approximations and in that case don’t try to do anything funky, sticks with the basics stuff