I am stuck on a simple problem and would highly appreciate your opinion. I have a optimization problem over $x$ with the objective function $$F=aG(x,y)+ (1-G(x,y))(1-x)$$ So the first order condition (value function) is given by $$V=a\frac{\partial G}{\partial x} - \frac{\partial G}{\partial x}(1-x)-G=0$$ I am looking for the partial derivative of $G$ with respect to $y$ at the $y^*$ which satisfies the first order condition. If I mechanically apply the implicit function theorem I get $$\frac{\partial V}{\partial y}=\frac{\partial^2 G}{\partial x\partial y}(a-1+x)-\frac{\partial G}{\partial y} \ \ \ \text{ and } \ \ \ \ \frac{\partial V}{\partial G}=-1 \\ \frac{\partial G}{\partial y}=- \frac{\partial V/\partial y}{\partial V /\partial G} $$ I am, however, unsure if I can simply take the derivative of $V$ with respect to $G$ since $\partial G / \partial x$ is also an implicit function of $G$, right?
I'm helplessly stuck, so if you can point if I need to go into a different direction that would be highly appreciated. Many thanks!