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
 A: For functions of one variable, I have never seen a problem, and wouldn't hesitate to treat them as fractions (multiplicatively). However, suppose you have $F(x,y)$ which implicitly defines a function $y=f(x)$, then
$\dfrac{dy}{dx} = -\dfrac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}$
If you just straight cancel as fractions, you'd get the wrong sign.
Edit: I just thought; there is a problem with the notation used which makes this error possible. The $\partial F$'s are different! One is given constant $x$ and the other constant $y$, hence they shouldn't necessarily cancel as they do in fractions. I guess it's rather pretty how they do manage to cancel to give a $-1$, but this particular case as just one instance, it's entirely possible for one symbol to represent different things in the same expression, so one would have to be much more careful about cancelling terms.   
A: 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.
A: The problem comes when you hit the second-order differentials.  When using second-order differentials in the standard notation, they DO NOT cancel.  
Example: Let's say that $y = x^3$ and $x = t^2$.  Find the second derivative of the first one:
$$y = x^3 \\
\frac{dy}{dx} = 3x^2 \\
\frac{d^2y}{dx^2} = 6x$$
Now, take the derivative of $x = t^2$.  You get:
$$\frac{dx}{dt} = 2t$$.
Now, let's try to cancel by squaring the second one:
$$\frac{d^2y}{dx^2}\cdot\left(\frac{dx}{dt}\right)^2 = 6x\cdot (2t)^2 \\
\frac{d^2y}{dx^2}\cdot\frac{dx^2}{dt^2} = 6(t^2) (2t^2) \\
\frac{d^2y}{dt^2} = 24t^4$$
The problem?  Let's do it again, but substitute at the beginning.  Since $y = x^3$ and $x = t^2$, that means that $y = (t^2)^3 = t^6$.  What's the second derivative?
$$\frac{dy}{dt} = 6t^5 \\
\frac{d^2y}{dt^2} = 30t^4$$
Uh oh!  Contradiction!
Now, the problem isn't with the concept of treating differentials as fractions, but our implementation of it. See Extending the Algebraic Manipulability of Differentials.  When you take the derivative of the derivative, if you are treating $\frac{dy}{dx}$ as a fraction, then the derivative of that fraction should use the quotient rule.  Therefore, the notation of the second derivative should be:
$$\frac{d^2y}{dx^2} - \frac{dy}{dx}\frac{d^2x}{dx^2}$$
Using this notation, then the higher-order differentials become algebraically manipulable again.  I could demonstrate it for you, but it is pretty messy.  See the paper if you want to see it happen. 
