real analysis – Can the derivative difference be arbitrarily larger than the function difference?

Background:

Let real numbers $x>y$,

$f:mathbb R^2tomathbb R$ such that $f(x,y)>0$ and $f(x,x)=0$

$g$ is an increasing real function.

Hypothesis: For any $f$, there exists a $g$ such that: $g'(x)-g'(y)geq f(x,y)(g(x)-g(y))$.

Is it possible to prove this hypothesis?

Motivation: Since $f(x,y)$ can be arbitrarily large, this question translates to: Can the derivative difference be arbitrarily larger than the function difference? From English the answer seems to be obviously no, but looking at the mathematical formula, the existence seems also obvious.

Possibly related: Solve for $f$ in an ODE involving two points: $f'(x)-f'(y)geq g(x,y)$ where $g$ is a known function.