I have two independent, continuous functions that satisfy the following inequality $$f(x,y) \ge g(x,y).$$ Now suppose $g(x,y)$ is set to be fixed, then $x$ and $y$ are constrained. Both functions are positive definite and their domains only take positive values.
My question: Is there any way to find an inequality (or any relation) between $\left. \frac{\partial f}{\partial x} \right|_y$ and $\left. \frac{\partial f}{\partial x} \right|_g$ ?
Or, is it possible to have an estimate of the changes of $f$, I mean $\Delta f = f(x_2,y_2)-f(x_1,y_1)$, when $g$ is kept fixed?