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)


Then use the "chain rule".  If x= y/2, so that y= 2x then for any function f(x), we have $\frac{df}{dy}= \frac{df}{dx}\frac{dy}{dx}= 2\frac{df}{dx}$. 
