The image of the assigment in question. (German)
I currently need to understand the correlation of the Big O notation and derivatives.
Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a $C^4$-Function and $(x,y) \in \mathbb{R}^2$ are set. We define for $h<0$
$$D_hf(x,y) := \sum_{i=-1}^1 \alpha_if(x+ih,y)$$ $$\tilde{D}_hf(x,y) := \frac{f(x+h,y+h)-f(x+h,y-h)-f(x-h,y+h)-f(x-h,y-h)}{4h^2}$$
I need to find the coefficient $\alpha_i$ so that $D_hf(x,y) = \frac{\partial}{\partial x}f(x,y) + O(h^2)$
As well as trying to prove that $\tilde{D}_hf(x,y) = \frac{\partial^2}{\partial x \partial y}f(x,y) + O(h^2)$
Currently i do not know how to go about this. I would probably need more understanding of Derivatives as well as the Big O notation. Any help would be appreciated to make me understand this. If i would have a point of reference to start out with.