In the same way as we derived central difference formulas for the first and second derivatives, we can also find difference formulas for the third and fourth derivatives of a 'neat' function \(f(x)\). Now we always need five points; for example, \[f'''(x_0)\approx\frac{f(x_0+2h)-2f(x_0+h)+2f(x_0-h)-f(x_0-2h)}{2h^3}\]

The table below summarises the central difference formulas in terms of the coefficients that can be used in a linear filter to calculate the derivative.

\[\begin{array}{|c|c|c|c|c|c|} \hline & f(x_0-2h) & f(x_0-h) & f(x_0) & f(x_0+h) & f(x_0+2h)\\ \hline 2hf'(x_0) & & -1 & 0 & 1 & \\ \hline h^2f''(x_0) & & 1 & -2 & 1 & \\ \hline 2h^3f'''(x_0) & -1 & 2 & 0 & -2 & 1\\ \hline h^4f''''(x) & 1 & -4 & 6 & -4 & 1 \\ \hline \end{array}\]