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Approximating of Higher Order Derivatives¶

It also possible to use Taylor series to approximate higher order derivatives (e.g., $$f''(x_j), f'''(x_j)$$, etc.). For example, taking the Taylor series around $$a = x_j$$ and then computing it at $$x = x_{j-1}$$ and $$x_{j+1}$$ gives

$f(x_{j-1}) = f(x_j) - hf^{\prime}(x_j) + \frac{h^2f''(x_j)}{2} - \frac{h^3f'''(x_j)}{6} + \cdots$

and

$f(x_{j+1}) = f(x_j) + hf^{\prime}(x_j) + \frac{h^2f''(x_j)}{2} + \frac{h^3f'''(x_j)}{6} + \cdots.$

If we add these two equations together, we get

$f(x_{j-1}) + f(x_{j+1}) = 2f(x_j) + h^2f''(x_j) + \frac{h^4f''''(x_j)}{24} + \cdots,$

and with some rearrangement gives the approximation $$$f''(x_j) \approx \frac{f(x_{j+1}) - 2f(x_j) + f(x_{j-1})}{h^2},$$$$and is$$O(h^2)$.