Existence of implicit and explicit derivatives

Question

Say we have an implicit function $y(x)$ defined by $f(x,y)=0$. Does the existence of $dy/dx$ on some region of $x$ imply the existence of $\partial f/\partial x$ and $\partial f/\partial y$ at $(x,y(x))$? If not, is there any expression we can form for $dy/dx$?

Answer 1

The answer is no. Consider the function $ f(x,y) = | x - y |.$ The set of points where $f = 0$ is precisely the set of points where $y = x$. The function $y(x) = x$ is differentiable, with derivative $1$ for all $x$. Yet $f$ itself does not have well-defined partial derivatives at any point of the form $(x,y(x))$.