Proving $f' = - \frac{\partial{F}/\partial{x}}{\partial{F}/\partial{y}}$
Denoting $g(x) = F(x, f(x))$, we have $g(x) = 0$ so $g'(x) = 0$.
$g'(x) = \partial{F}/\partial{x} + f'(x) \cdot \partial{F}/\partial{y} = 0$ which concludes
Denoting $g(x) = F(x, f(x))$, we have $g(x) = 0$ so $g'(x) = 0$.
$g'(x) = \partial{F}/\partial{x} + f'(x) \cdot \partial{F}/\partial{y} = 0$ which concludes