When can't $dy/dx$ be used as a ratio/fraction? 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

I do not know very much. I am still studying like you. One thing I know hope this would help and I might be wrong.   
If we consider $\dfrac{dy}{dx}$ as a fraction and say $\dfrac{dy}{dx} = \dfrac{1}{\dfrac{dx}{dy}}$ because it is a fraction then there is a problem.
Consider $y=f(x)=x^0$. Now $\dfrac{dy}{dx}=0$ but $\dfrac{dx}{dy}$ is undefined  so we see that $\dfrac{dy}{dx} \neq \dfrac{1}{\dfrac{dx}{dy}}$.  
Currently I am reading this book for calculus. The author has explained chain rule very good so you might like it.  
I had somewhere read that for chain rule to work $x$ and $y$ should be expressible in form $\phi(x,y)=0$. - I might be wrong.  
I had also read somewhere that $y=f(x)$ should be a bijective function. $-$ Again I might be wrong here.
