If the equations $f(x, y, u, v) = 0$ and $g(x, y, u, v) = 0$ can be solved for $u$ and $v$ as differentiable functions of $x$ and $y$, compute their first partial derivatives.
Pretty lost on this one. Think it might have to do with Newton's method. Any help would be appreciated.
Write the equations as $$ 0=f(x,y,u(x,y),v(x,y))\text{ and }0=g(x,y,u(x,y),v(x,y)) $$ compute partial derivatives for $x$ (analogous for $y$) using the chain rule and identify and solve the resulting linear system for the partial derivatives of $u$ and $v$.
See also "Implicit differentiation".