Numerical Differentiation for 2nd Derivative

For second derivatives $f^{\prime \prime }(x)$, calculating the change in slope requires 3 points:

$$\begin{array}{rl}f(x_{i+2})& =f(x_i)+f^{\prime }(x_i)(2h)+\frac{f^{\prime \prime }(x_i)}{2!}(2h{)}^2+\dots \\ f(x_{i+1})& =f(x_i)+f^{\prime }(x_i)h+\frac{f^{\prime \prime }(x_i)}{2!}{h}^2+\dots \end{array}$$

If we subtract subtract the equations as $(1)-2(2)$:

$$f(x_{i+2})-2f(x_{i+1})=-f(x_i)+f^{\prime \prime }(x_i)h^2+\dots$$

Solving for $f^{\prime \prime }(x_i)$ gives:

$$\begin{array}{rl}\text{Forward:}& f^{\prime \prime }(x_i)=\frac{f(x_{i+2})-2f(x_{i+1})+f(x_i)}{h^2}+O(h)\\ \text{Backward:}& f^{\prime \prime }(x_i)=\frac{f(x_i)-2f(x_{i-1})+f(x_i)}{h^2}+O(h)\\ \text{Centered:}& f^{\prime \prime }(x_i)=\frac{f(x_{i+1})-2f(x_i)+f(x_{i-1})}{h^2}+O(h)\end{array}$$

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