By searching this question, I found this: Can I ever go wrong if I keep thinking of derivatives as ratios?
However, the answers don't have what I'm looking for! (Edit: Meaning, a counterexample. There is one involving partial derivatives, but then the only difference has to do with signs, which means that $dy/dx$ can still be interpreted as a ratio. Thanks, fuglege)
So long as you treat $dx$ as $dx$ (meaning one "object", that is, not d times x), I still have yet to see an example where using the differentials as a fraction yields an incorrect answer (and I have watched almost all the khan academy videos on calculus and read quite a bit out of the ubiquitous Stewart calculus book!)
Note that I am not asking about non-standard analysis. I have found an online textbook and I am currently reading it. I am only trying to find a counterexample to using the differentials as a fraction.
Thanks very much