- #1

- 54

- 0

## Main Question or Discussion Point

**Problem**- Find backward finite difference approximations to first, second and third order derivatives to error of order h^3

**Attempt**

By Tailor’s series expansion

f(x-h) = f(x) - h f’(x) + h^2/2! f’’(x) - h^3/3! f’’’(x) + …

Therefore, f’(x) with error of order h^3 is given by

f(x-h) = f(x) - h f’(x) + h^2/2! f’’(x) - h^3/3! f’’’(x)

h f’(x) = f(x) – f(x-h) + h^2/2! f’’(x) – h^3/3! f’’’(x)

f’(x) = [f(x) – f(x-h)]/h + h/2! f’’(x) – h^2/3! f’’’(x)

But I do not know what to substitute for f''(x) and f'''(x)

Also, what to do for 2nd and 3rd order derivatives?