if $x = 2y$ what does it mean to say that $\frac{d}{dx} = \frac{1}{2}\frac{d}{dy}$? If $x = 2y$ what does it mean to say that $\frac{d}{dx} = \frac{1}{2}\frac{d}{dy}$?
The question is the following: let $f$ be differentiable (smooth if necessary). 
I believe that we could try to interpret the above  relation the following way:
Write $$f(x) = g(y) $$
$$g(y) = f(2y) $$
$$\frac{d}{dy} g(y) = f'(2y)(2y)' = 2 f'(x)  = 2\frac{d}{dx}f(x)$$
This seems too euristic for me.
How can we understand  better sentences like
$$dy = f'(x)dx  \Rightarrow \frac{d}{dy} = \frac{d}{dx}\frac{1}{f'(x)}$$
Just to add a context, the question is issued from the following lines of Ethier and Kurz (Markov processes characterization and convergence pg 366)


$$x=2y \implies \frac{dx}{dy}=2 \implies \frac{dy}{dx} = \frac 12 \implies \frac{1}{dx} = \frac 12 \frac{1}{dy}$$
Note: Here we applied to $\frac{dx}{dy}$ rules as it would be a fraction. Although this is not allowed in classical analysis, this kind of argumentation is often used (for example in physics). But there are theories of analysis which allows calculations with infinitesimals (e.g. non-standard analysis). See Why it is absolutely mistaken to cancel out differentials? for more details on this issue.
