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Simplification and implicit function

I have two question in one : 1) I have this $z= \frac{\alpha x^{\alpha-1}y^\beta y^{-\beta-1}}{\beta x^\alpha}$. There are obvious simplification since this is equal to $\frac{\alpha}{\beta} \frac{y}{x}$.

z.full_simplify() has no impact z.expand() simplify on $y$ but not on $x$

So what can I do to obtain the good result.

2) how to take the result and consider it as an implicit function to obtain the derivative de $y$. as a function of x