I have seen how people implicitly differentiate the equation $x^2 + y^2 = c$.
$$d/dx(x^2) + d/dx(y^2) = d/dx(c)$$
treating "$y$" as "$f(x)$" and using the chainrule we get
$$2x + 2y(y') = 0$$
and solving for $y'$
The problem is that I just don´t understand implicit differentiation, I do know the rules but they don´t make any sense to me. The fact that it is valid to differentiate both "$x$" and "$y$" on the same side of the equation is what´s bothering me and even if I see "$y$" as a function of "$x$" I just end up imagining
$$x^2 + (-x^2 + c) = c$$
which doesn´t help me. I also don´t know very much about partial derivatives but I´m willing to learn about them if that helps me understand implicit differentiation.
I really appreciate any thoughts or ideas. Thank you!