I have derived the following first order condition
$$g(x,C'(x),..)=1,$$
where $x$ is the production level and $C(x)$ the cost of $x$, and $C'(x)$ is the first derivative of $C(x)$ (i.e. the marginal cost). By the implicit function theorem, we can derive how production changes with marginal cost:
$$\frac{\partial x}{\partial C'(x)}=-\frac{\partial g/\partial C'(x)}{\partial g/\partial x}$$
Where, the numerator on the RHS equals
$$\partial g/\partial C'(x)=\frac{\partial g/\partial x}{d C'(x)}$$
This then allows us to simplify the second equation
$$-\frac{\frac{\partial g/\partial x}{d C'(x)}}{\partial g/\partial x}=-\frac{1}{C''(x)}$$
And we then conclude that $\frac{\partial x}{\partial C'(x)}=-\frac{1}{C''(x)}$. This is clearly absurd, since with a concave cost function, the derivative of $x$ with respect to marginal cost will always be positive (i.e. production is increasing in marginal cost). What mistake did I make to derive something this absurd?
The step I think may be problematic is the third equation $$\partial g/\partial C'(x)=\frac{\partial g/\partial x}{d C'(x)}.$$ I asked about this elsewhere, and the step seemed fine to some.
Your problem comes from assuming that the cost function is concave.
With perfectly competitive markets, the firm’s profit maximisation problem cannot be solved using first-order conditions. With concave costs, the second derivative of the profit function will be convex, so you can’t solve it via first-order conditions. Attempting to do so gives you a solution that minimises profits. (This might also explain your nonsensical result.)
Perhaps you intended to assume that production costs are convex. This would give you the intuitive result that output is decreasing in marginal cost.