error bound of $I=\int_{0}^{1}\left(f'(x)\right)^{2}dx\thickapprox h\sum_{i=1}^{N}\left(\frac{f(ih)-f((i-1)h)}{h}\right)^{2}$

Question

let for $f\in c^{3}$ and $I=\int_{0}^{1}\left(f'(x)\right)^{2}dx\thickapprox h\sum_{i=1}^{N}\left(\frac{f(ih)-f((i-1)h)}{h}\right)^{2}$. whereas $h=\frac{1}{N}$ find the maximal error bound in [0,1]. meaning find a term depends only on f derivatives. what is the algebric accuracy of the bound? My try: I tried to write f as interpolation of f(ih), and take the derivative, I got very nasty terms. Does someone has an intelligent solution? The following question might be helpful to find the bound: Prove that $\int_a^b (f'(x))^2dx - h\sum_{i=0}^{n-1} \left(\frac{f(x_{i+1})-f(x_i)}{h}\right)^2 = O(h^2) $