# Multivariate finite difference formulas

Consider a Taylor expansion of a function $$f$$ of $$N$$ variables $$\mathbf{x}$$, about $$\mathbf{x}=\mathbf{0}$$:

$$f(\mathbf{x}) = f(\mathbf{0})+\sum_i^Nc_ix_i + \frac{1}{2!} \sum_{i,j}^N c_{ij}x_ix_j + \frac{1}{3!} \sum_{i,j,k}^N c_{ijk}x_ix_jx_k + \frac{1}{4!} \sum_{i,j,k,l}^N c_{ijkl}x_ix_jx_kx_l + \cdots$$

where indices $$ijkl$$ represent different variables, $$c_{ij}$$ are unknown constants representing 2nd derivatives with respect to $$x_i$$ and $$x_j$$, $$c_{ijk}$$ are unknown constants representing 3rd derivatives, etc.

Where can I find finite difference equations for the constants $$c$$ (i.e., the derivatives), at least up to 4th order?

Furthermore, how can I derive finite difference equations for these constants?