Yes, and what is aimed for is separation of variables to find a solution of this differential equation. (The differential operator d should be printed as upright letter btw)
You can't multiply both sides by dx in much the same way you can't differentiate a duck. That said, even pure mathematicians sort of think of it that way as a useful shorthand.
dy/dx is the same as (y1 - y2) / (x1 - x2) as the distance between the two points approaches zero. “dx” and “dy” aren’t very useful measurements on their own though.
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