the eccentric derivative

When I first studied ordinary, single-variable derivatives, they were relatively simple and the rules weren't so confusing. In fact, many of them belonged to the "common sense" group. Take, for example:
dy/dx = (dy/du)(du/dx)
If you treat "d(...)" as an operator and as a single entity, and manipulate derivatives as if they are actual fractions, then the chain rule appears to be "obvious".

I can't quite say the same thing for partial derivatives, however. Some partial derivative rules are just... strange:
dy/dx = -(∂z/∂x)/(∂z/∂y), if z(x, y) = 0.
The surprising part of this equation is the negative sign. While I can derive the equation itself without issues, it appears, at least to me, quite surprising that partial derivatives don't behave the way total derivatives do. Now I realize that the distinction is indeed necessary, and this difference in behavior and rules is one reason.

Take for example:
dy/dx = 1/(dx/dy)
This is always true, but it isn't true if I replaced it with partial derivatives. Instead, the inverse becomes a "sort-of" matrix inverse, and things are more complicated. Put simply, there is a way to invert partial derivatives, but the formula is far more complicated than what is shown above for total derivatives.

There is often a common trend in math: as things become more complicated, involving "higher-order" entities, some of the simple rules that originally applied would breakdown, replaced by more general, but complicated rules. One might call this unruly, but I would consider this "elegant" for most cases. E.g. Stoke's theorem is more complicated but far more general than the single-variable fundamental theorem of calculus.

It's time like these that remind me why I love math in the first place.