AFAIK, partial derivatives made it to the forefront as a result of coming up with an equation for a vibrating string i.e., 1D wave equation.
From purely a physical phenomena to math mapping POV: Why does a partial derivative show up? That is, how do we justify that we need something like a partial derivative to address something that ordinary derivatives are deficient at?
Let's assume that the vibrating string was the genesis of partial differentials (AFAIK Bernoulli (one of them) had invented this concept sooner but let's assume otherwise for now). The question then is what aspect of capturing the vibrating string problem's as math equations leads one to "invent" partial derivatives?
I've been pondering over this for a while ever since I learned about the history and derivation of the 1D wave equation. I don't think I understand the "need for partial derivates" well enough to answer this question.
PS: Decided to post in the Physics.SE vs. Math.SE since I'd like to understand PDEs from the POV of physical phenomena and build intuition about their need.