What is the difference between $\mathbb{C}[x]_{(x)}[y]$ and $\mathbb{C}[x,y]_{(x)}$? To me they are both equal to:
$\{ \frac{f(x, y)}{g(x,y)} | g(0, y) \neq 0 \}$
Is this true and if it isn't can anyone provide any intuition as to why it isn't, please.