Let $f: \mathbb R^2 \rightarrow \mathbb R$ be a smooth strictly convex function with unique minimum at $0$ such that all level sets $A_x:=\left\{z ; f(z) \le x \right\}$ are compact. Imagine something like $f(z)=\Vert z \Vert^2.$

Define the integral function

$$F(x):=\int_{A_x} g(z) dz$$

where $g$ is as smooth as you like.

Question: Is there a way to analytically determine an expression for $F''(x)$?