Solution
= Solution
The planar <implicit function theorem> states that if $F$ is $C^1$ near $(a,b)$, $F(a,b)=0$, and $F_y(a,b)\neq0$, then in some product <neighbourhood> the equation $F(x,y)=0$ is equivalent to $y=h(x)$ for a unique $C^1$ function with $h(a)=b$. Moreover,
$$
\boxed{h'(x)=-\frac{F_x(x,h(x))}{F_y(x,h(x))}.}
$$
If instead $F_x(a,b)\neq0$, exchange the coordinates to solve locally for $x$ as a function of $y$.