The meaning of $\frac{dy}{dx}$ for an implicit curve $F(x, y) = c$ at a point which is not at the curve

Question

What is the meaning of $\frac{dy}{dx}$ for $$xy^2 + x^2 - \frac{y}{x} = 2$$ at the point $(1,1)$ which is not at the curve? I know if the point is on the curve the derivative is the slope of the tangent at that point.

Answer 1

The expression on the l.h.s. of the equation is of the form $$F(x, y) = c,$$ and differentiating gives $$\frac{\partial F}{\partial x} + \frac{\partial F}{\partial y} \frac{dy}{dx} = 0 .$$ In particular, $c$ plays no role, so we can interpret $$\left.\frac{dy}{dx}\right\vert_{(x_0, y_0)}$$ as the slope of the tangent line to the unique level curve of $F(x, y)$ that passes through $(x_0, y_0)$, namely, the curve $$F(x, y) = F(x_0, y_0) .$$ (This statement comes with the usual caveats about differentiability and degeneracy of curves.)