I have been researching the history of calculus (which means watching various math youtubes) and thinking about differential arguments involving $dx$ and $dy$. I noticed that in the quotient rule for differentials we cannot so easily ignore a term consistently. $$ d\left(\frac{x}{y}\right) = \frac{x + dx}{y+dy}- \frac{x}{y} = \frac{y (x + dx) - x (y + dy)}{y (y + dy)}= \frac{y~dx - x~dy }{y^2 + \color{red}{y~dy}} $$
Why do we ignore the $\color{red}{y~dy}$ term in the denominator, but not the $y~dx$ , $x ~ dy$ terms in the numerator? We see from the numerator that we ought not to ignore products of variables and differentials, otherwise we would obtain $ 0/y^2 = 0$ . I can understand ignoring the product of two differentials, which comes up in the derivation of the differential product rule. viz., $d(x\cdot y) = (x + \Delta x )(y + \Delta y) - x~ y= x ~ \Delta y + y ~ \Delta x + \Delta x ~ \Delta y $. It makes sense to ignore the last term, $\Delta x ~ \Delta y$, because it is a second order vanishingly small zero-approaching quantity. So we could say first order behavior dominates. Does $y^2$ dominate over $y ~dy$ in the quotient?
\begin{align} d\left(\frac{x}{y}\right) &= \frac{y~dx - x~dy }{y^2 + ydy} \\ &= \left( y~dx - x~dy \right) \left( \frac{1}{y^2} + O(dy) \right) \\ &= \frac{y~dx - x~dy}{y^2} + O(d^2) \end{align}
where I indicated with $O(d^2)$ terms of second order or higher.