f (x) = f (x_{0}) + (x − x_{0}) f′|_{x_{0}} + \frac{1}{2} (x − x_{0})^{2} f′′|_{x_{0}} + \frac{1}{3!} (x − x_{0})^{3} f′′′|_{x_{0}} + …

(1)

f (x_{0} ± dx) = f (x_{0}) ± dxf′|_{x_{0}} + \frac{1}{2}dx^{2}f′′|_{x_{0}} ± \frac{1}{3!}dx^{3}f′′′|_{x_{0}} + … (2)

f_{i+1} − f_{i} = dxf′|_{x_{i}} + \frac{1}{2}dx^{2}f′′|_{x_{i}} + \frac{1}{3!}dx^{3}f′′′|_{x_{i}} + …  (3)

f′|_{x_{i}} = \frac{f_{i+1} − f_{i}}{dx} − \frac{1}{2}dxf′′|_{x_{i}} + …  (4)

f′|_{x_{i}} = \frac{f_{i} − f_{i−1}}{dx} + \frac{1}{2}dxf′′|_{x_{i}} + …  (5)

f_{i+1} − f_{i−1} = 2dxf′|_{x_{i}} + \frac{2}{3!}dx^{3}f′′′|_{x_{i}} + …  (6)

f′|_{x_{i}} = \frac{f_{i+1} − f_{i−1}}{2dx} − \frac{2}{3!}dx^{2}f′′′|_{x_{i}} + …  (7)

f_{1} = f_{0} + dxf′|_{x_{0}} + \frac{1}{2}dx^{2}f′′|_{x_{0}} + \frac{1}{6}dx^{3}f′′′|_{x_{0}} + …  (8)

f_{2} = f_{0} + 2dxf′|_{x_{0}} + 2dx^{2}f′′|_{x_{0}} + \frac{8}{6}dx^{3}f′′′|_{x_{0}} + …  (9)

4f_{1} − f_{2} = 3f_{0} + 2dxf′|_{x_{0}} − \frac{2}{3}dx^{3}f′′′|_{x_{0}} + …  (10)

f′|_{x_{0}} = \frac{4f_{1} − 3f_{0} − f_{2}}{2dx} − \frac{1}{3}dx^{2}f′′′|_{x_{0}} + …  (11)

f_{i+1} + f_{i−1} = 2f_{i} + dx^{2}f′′|_{x_{i}} + \frac{1}{12}dx^{4}f′′′′|_{x_{i}}  (12)

f′′|_{x_{i}} = \frac{f_{i+1} − 2f_{i} + f_{i−1}}{dx^{2}} − \frac{1}{12}dx^{2}f′′′′|_{x_{i}} + …  (13)