I need some help with some implicit differentiation.

I have a system where $\displaystyle {\color{red}x}$ in a choice variable that someone gets to pick. For each choice of $\displaystyle {\color{red}x}$ the following non-linear system of equations gets solved for $\displaystyle {\color{red}y}$ and $\displaystyle {\color{red}z}$.
$\displaystyle F(x,y,z)&=&0\\G(x,y,z)&=&0$
and spits out a unique $\displaystyle (x,y(x),z(x))$. This $\displaystyle (x,y(x),z(x))$ is then used to evaluate the expression $\displaystyle H(x,y,z)$. Now I am writing $\displaystyle {\color{red}y(x)}$ and$\displaystyle {\color{red}z(x)}$ but $\displaystyle {\color{red}y}$ and $\displaystyle {\color{red}z} $ cannot be expressed in terms of elementary functions of $\displaystyle {\color{red}x}$ since the above system of equations cannot be solved analytically.

The idea is to find an $\displaystyle {\color{red}x}$ where $\displaystyle \frac{d H(x,y(x),z(x))}{dx} =0$. I don't actually want to solve this problem right away in the sense that I don't need the $\displaystyle {\color{red}x}$ that solves the first order condition, rather I need the expression of the first order condition itself which I need to plug in somewhere else. That's why I am not using Lagrange multipliers here.

Any help understanding the math here at a conceptual level and making progress on this problem will be greatly appreciated.