if a function depends only on multiple variables then those are partial derivatives, if the function depends only on one variable then there is only one derivative d/dx
I get that, but ultimately, why does the notation distinguish between the two classes "function with zero arguments held constant" and "function with n >= 1 arguments held constant"? It seems somewhat of an arbitrary distinction, when in fact the value of n is always clear from the context.
I think that the difference is from historical reasons, anyway when if you introduce in school partial derivatives with functions of only one variable that is confusing because here partial is not partial but global.
I think his point is that you can always use the partial derivative notation and be correct.