Hello guys I want to make sure myself in determming when is $y$ as function of $x$,so for this, let us consider following question. If the equation of circle is given by this
$x^2+y^2=25$ and question is find the equation of tangent of circle at point $(3,4)$,then it is clear that we need calculate the derivative and also we can express $ y$ as
$y=\sqrt{25-x^2}$ but, let us consider following situation
$x^3+y^3=6\cdot x \cdot y$
it's name is The Folium of Descartes and my question is the same : calculate the equation of the tangent of this folium at point $(3,4)$. I know that somehow expressing $y$ as a function of $x$ is difficult but could't we do it by some even long mathematical manipulation? So what is the strict explanation when is $y$ a function of $x$ and when is it not? I need to be educated in this topic and please explain me ways of this,thanks a lot.