Let $f$ be 3 times continuously differentiable. How do I show that
$f'(x) - {f(x+h) - f(x-h) \over 2h}\in O(h^2)$?

By the intermediate value theorem, I know that there exists a $\xi \in [x-h,x+h]: f'(\xi) =\frac{f(x+h) - f(x-h)}{2h}$. I think I could use Taylor but I doesn't lead me to any solution.