what is $d(y/x)$ in $ d\log(y/x)=\frac{d(y/x)}{(y/x)}$? For example what is meant by $d(y/x)$ in right hand numerator in the following?

$$ d\log(y/x)=\frac{d(y/x)}{(y/x)}$$

Is the same as $\frac{dy}{dx}$? If not, what does it mean?
No, it is not the same

$$
{\rm d}\left(\frac{y}{x}\right) = \frac{{\rm d}y}{x} - \frac{y}{x^2}{\rm d}x \not= \frac{{\rm d}y}{{\rm d}x}
$$
For any variable z, d(z) is a differential. Differentials are explained well in many textbooks and internet articles.