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I have two question in one : 1) I have this $z=\frac{αx^{α−1}y^βy^{−β−1}}{βx^α}$. There are obvious simplification since this is equal to $\frac{α}{β}\frac{y}{x}$.

It isn't:

sage: foo=a*x^(a-1)*y^b*y^-(b+1)/(b*x^a);foo
a*x^(a - 1)*y^b*y^(-b - 1)/(b*x^a)
sage: foo.canonicalize_radical()

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

sage: foo.canonicalize_radical().diff(y)

which should be obvious given the (correct) result of your simplification...