TITLE: What does implicit differentiation really mean
QUESTION [5 upvotes]: What does dy/dx really mean in the context of implicit differentiation
The problem is that I can't relate it back to the limit definition. 
If you can't write y explicitly as a function of x, often y approaches multiple values as x approach some value, or at some x, y has multiple values hence the limit does not exist and therefore the derivative will not exist

REPLY [3 votes]: For implicit differentiation you will have an equation that -- at least locally -- is satisfied by the points on a curve through the plane.
Imagine that we find a parameterization of the curve such that the set in question is
$$ \{ (f_x(t),f_y(t)) \mid t\in [0,1] \} $$
and otherwise "sufficiently well-behaved". Then at the point $(f_x(t_0),f_y(t_0))$ the implicit derivative ought to be the limit
$$ \lim_{h\to 0}\frac{f_y(t_0+h)-f_y(t_0)}{f_x(t_0+h)-f_x(t_0)} $$