estimation of different quotient of a non negative convex function

Question

Let $f$: $\mathbb R\to\mathbb R^+$ be a differentiable convex function and we recall the difference quotient is defined by $$ \Delta_hf(x):=\frac{f(x+h)-f(x)}h. $$ We have that $\lim_{h\to 0}\Delta_hf(x)=f'(x)$ as well as, since $f(x)$ is convex, that $$ \Delta_{h_1} f(x)\geq \Delta_{h_2}f(x)\text{ if }h_1>h_2. $$ However, could we have some sort of estimation on the value $$ |\Delta_{h} f(x)-f'(x)| $$ without reference to the point $x$?