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- Can I always treat derivatives as a small element lf y divided by a small element of x?

In physics, the differential is always treated as a little change or a tiny element of something, be it volume, area, etc. However, when I differentiate a function with respect to another, I always of it as a change divided by a change, not an element divided by an element: like when volume is an explicit function of area, I think of how much the volume changes, rather than taking a small element of volume and dividing it by a small element of area. My question is : are the two "definitions" equally valid? Can I always think of a derivative both ways? + I only know calculus and elementary differential equations so I will probably not understand any answer given in terms of differential forms, so please keep it simple