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

$2x+2y({y}^{\prime})=0$

and solving for ${y}^{\prime}$

${y}^{\prime}=-2x/2y$

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.

$d/dx({x}^{2})+d/dx({y}^{2})=d/dx(c)$

treating "$y$" as "$f(x)$" and using the chainrule we get

$2x+2y({y}^{\prime})=0$

and solving for ${y}^{\prime}$

${y}^{\prime}=-2x/2y$

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.