When not to treat dy/dx as a fraction in single-variable calculus? While I do know that $\frac{dy}{dx}$ isn't a fraction and shouldn't be treated as such, in many situations, doing things like multiplying both sides by $dx$ and integrating, cancelling terms, doing things like $\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}$ works out just fine.
So I wanted to know: Are there any particular cases (in single-variable calculus) we have to look out for, where treating $\frac{dy}{dx}$ as a fraction gives incorrect answers, in particular, at an introductory level?
Note: Please provide specific instances and examples where treating $\frac{dy}{dx}$ as a fraction fails

Treating derivatives as ratios allows us to come up with a heuristic argument for the product rule, if we agree that the product of two differentials is negligible:
\begin{align}
d(xy) &= (x+dx)(y+dy) - xy \\[5pt]
&=x \, dy+y\,dx+dx \, dy \\[5pt]
&= x \, dy+y \, dx \\[5pt]
\frac{d(xy)}{dt} &= x\frac{dy}{dt}+y\frac{dx}{dt}
\end{align}
However, try doing the same thing with the quotient rule, and things don't work out so nicely:
\begin{align}
d\left(\frac{x}{y}\right)&=\frac{x+dx}{y+dy}-\frac{x}{y} \\[5pt]
&= \frac{y(x+dx)-x(y+dy)}{y(y+dy)} \\[5pt]
&= \frac{y \, dx-x \, dy}{y^2+y \, dy} \, .
\end{align}
Here, it is difficult to argue why the term $y \, dy$ should be negligible, meaning that the quotient rule cannot be derived in a straightforward manner by treating derivatives as ratios.
