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.
