I understand the method used in implicit differentiation, it's just an application of the chain rule. But why can you define $y$ as a function of $x$?
In this equation for example:
$x^2 + y^2 = 1$
$2x + 2yy' = 0 $
Why isn't it just this?:
$2x + 2y = 0$
Thank you in advance.