TITLE: Can I ever go wrong if I keep thinking of derivatives as ratios?
QUESTION [27 upvotes]: I have been forewarned about it, I have read the answers here, but I haven't seen a counter example where it doesn't work. I know that it isnt really a fraction, but does it effectively get the same result in all cases, or are there counterexamples I must be warned about. 
I do not mean obvious counter examples, like $$\frac{dy}{dx}+\frac{du}{dv} = \frac{dydv+dxdu}{dx dv}$$ which as far as I know doesn't really mean anything.

REPLY [7 votes]: This question has come up at MathOverflow:
https://mathoverflow.net/questions/73492/how-misleading-is-it-to-regard-fracdydx-as-a-fraction
and overlaps with other questions on this site:
Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
If $\frac{dy}{dt}dt$ doesn't cancel, then what do you call it?
There are several interesting answers at each of those links.