$x$ dimension → discrete points in $x$ → $x_n, \, n = 0, \ldots, N+1$ with constant grid spacing $$ \Delta x = x_{n+1} - x_n $$ $N$ grid points, and two ‘ghost’ points to satisfy boundary conditions

Derivative of function $h$ at point $x_n$ is given by $$ \left. \frac{\partial h}{\partial x} \right|_{x=x_n} = \lim_{\Delta x \to 0} \frac{h(x_n + \Delta x) - h(x_n - \Delta x)}{2\Delta x} = \lim_{\Delta x \to 0} \frac{h_{n+1} - h_{n-1}}{2\Delta x} $$ with $h_n = h(x_n), \, h_{n-1} = h(x_{n-1}), \, h_{n+1} = h(x_{n+1})$

Stop the limiting process at finite $\Delta x \Rightarrow$ finite difference $$ \left. \frac{\partial h}{\partial x} \right|_{x=x_n} \approx \frac{h_{n+1} - h_{n-1}}{2\Delta x} $$ That would be a central difference approximation to $\partial h / \partial x$

Forward difference $$ \left. \frac{\partial h}{\partial x} \right|_{x=x_n} \approx \frac{h_{n+1} - h_n}{\Delta x} \equiv \delta^{+}(h_n) $$

Backward difference $$ \left. \frac{\partial h}{\partial x} \right|_{x=x_n} \approx \frac{h_n - h_{n-1}}{\Delta x} \equiv \delta^{-}(h_n) $$ with finite differencing operators $\delta^{+}(h_n)$ and $\delta^{-}(h_n)$