# divided difference involving derivatives

$$f(x)= (x+1)^4$$

points where function is measured/sampled are 0 and 1. I thought when we have two conservative derivative on same point$$\ x$$ as $$f'(x)$$, in next column their divided difference becomes $$f''(x)$$

so ,in the image, why then we have $$\frac{1}{2}f''(0)$$ instead of just $$f''(0)$$

$$f[x,x+h,x+2h]=\frac{f(x+2h)-2f(x+h)+f(x)}{2h^2}.$$
If $$f$$ is twice differentiable at $$x$$, then this difference tends to $$\frac{1}{2}f^{''}(x),$$ whenever $$h\to 0$$.