fundamental question about alternate "differential" style calculus

This is going to be a frustrating question - I'm not even sure I can
explain the question clearly. Sorry. I'll try my best.

Several times I've come across an alternate style of differential
calculus, where instead of saying d/dx(...), the "division" by "dx"
is discarded, giving rules such as
d(uv) = (du)v + u(dv)
dx^3 = 3x^2 dx
These rules can (apparently) be derived by just multiplying the "d/dx"
version of the rule by "dx". Then, lastly, when the equation is
in the form "A dx", one can drop the dx (effectively "dividing" by it
to compensate for the earlier multiplication), obtaining the derivative,
(which can then be set to zero, etc.).
"3x^2 dx" from the second rule above is of that form, so can drop the dx,
giving
derivative = 3x^2

Ok, so now towards the question: suppose we want to find x that best minimizes
(y-ax)^2 = (y-ax)(y-ax)
Applying the first rule above,
d(y-ax)(y-ax) + (y-ax)d(y-ax)
= 2(y-ax)d(y-ax)

So, what is d(y-ax)? I know very well it "should" be -a, BUT,
the d() notation doesn't "know" that the derivative will be
with respect to x yet! It seems like the differentital d(y-ax)
should be the differential of all these quantities y,a,x as
combined in the given form y-ax, and that there should be some
further bookkeeping or evidence that dy=0 for example.

Suppose the equation was a bit more complicated,
d(pq^2 +r^3)
how do we go about knowing which of these are zero and which are not?

That's the best I can do. This is more of a philosophy or
"how to think about things" question than a core math question.

Thanks for any thoughs or discussion.
.

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