Numerical Differentiation

Tables for Derivatives on Uniform Grids

Some First Derivative Expressions for a Uniform Mesh

Derivative at point $i$	Discrete Representation (uniform mesh)	Order
Forward Difference	$\left. \frac{\mathrm{d}\phi}{\mathrm{d}x} \right|_{i} \approx \frac{\phi_{i+1}-\phi_{i}}{\Delta x}$	$\mathcal{O}\left(\Delta x \right)$
Backward Difference	$\left. \frac{\mathrm{d}}{\mathrm{d}x} \right|_{i} \approx \frac{\phi_{i}-\phi_{i-1}}{\Delta x}$	$\mathcal{O}\left(\Delta x \right)$
Central Difference	$\left. \frac{\mathrm{d}}{\mathrm{d}x} \right|_{i} \approx\frac{\mathrm{d}}{\mathrm{d}x} \approx \frac{\phi_{i+1}-\phi_{i-1}}{2\Delta x}$	$\mathcal{O}\left(\Delta x^2 \right)$

Some Second Derivative Expressions for a Uniform Mesh

Derivative at point $i$	Discrete Representation (uniform mesh)	Order