I need some help with some implicit differentiation.

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

The idea is to find an {\color{red}x} where \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 {\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.