Intuition behind calculus notation I have read somewhere sometime ago (very specific, I know...), that it would not be correct to treat rate of
change, i.e. any arbitrary $\frac{dy}{dx} \ $ as a fraction, and thus it is not possible to perform any algebraic manipulation with it. I should have questioned that when I have read it in the first place, I would not have been asking this question now then.
Anyway.. I have seen this "rule" being ignored many many times, e.g. $\displaystyle{\frac{dy}{dx} \equiv \frac{1}{\frac{dx}{dy}}} \ $ or $ \int a(y) \frac{dy}{dx} dx \equiv \int a(y) dy$. I don't understand how you can cancel $dx$ in the integral either, isn't it just a notation to say what to integrate with respect to? 
I want to understand how is this possible. Thank you! 
 A: It's not just notation. Differentiation has almost all the properties of a true fraction, but you have to keep in mind that differentiation is actually an operator (things get "hairy" when you go into multiple dimensions, where partial and total differentials are not the same thing, and if the coordinates are general curvilinear you get into trouble and have to introduce metrics). This notation gets justified and formalized in the calculus of differential forms, which you should probably read about.
Also, the derivative can be interpreted as a limit of a quotient of small increments in both directions (a physicist's view on calculus), and can be treated as a fraction when the limit of an entire expression can be taken together.
$$f'(x)=\lim_{dx\to 0}\frac{f(x+dx)-f(x)}{dx}$$
So you see, the notation of fractions is not a problem: the main problem is what $\rm d$ is. If it's treated as a small increment and the limit is performed, then things work well as long as this interpretation of a differential holds (most 1D constructions will work just fine). If it's a differential form, a member of a vector space that generates coordinates, then you will see a deeper structure that actually ties differential calculus to the metrics of space. If you study general relativity, this is the only way to survive. It actually makes things clearer: you get duality of covariant and contravariant vectors and much more. But that approach is difficult, as it requires a completely different perspective on differentiation.
