Numerical Methods/Numerical Differentiation

Often in Physics or Engineering it is necessary to use a calculus operation known as differentiation. Unlike textbook mathematics, the differentiated functions are data generated by an experiment or a computer code.

Begin with the Taylor series as seen in Equation 1.

$f(x+h)=f(x)+f^{\text{'}}(x)h+\frac{f^{(2)}(x)}{2!}{h}^2+\frac{f^{(3)}(x)}{3!}{h}^3+\cdots (1)$

Next by cutting off the Taylor series after the fourth term and evaluating it at h and -h yields Equations (2) and (3).

$f(x+h)=f(x)+f^{\text{'}}(x)h+\frac{f^{(2)}(x)}{2!}{h}^2+\frac{f^{(3)}(c_1)}{3!}{h}^3(2)$

$f(x-h)=f(x)-f^{\text{'}}(x)h+\frac{f^{(2)}(x)}{2!}{h}^2-\frac{f^{(3)}(c_2)}{3!}{h}^3(3)$

Then by subtracting Equation (2) by Equation (3) yields.

$f(x+h)-f(x-h)=2f^{\text{'}}(x)h+\frac{f^{(3)}(c_1)}{3!}{h}^3+\frac{f^{(3)}(c_2)}{3!}{h}^3$

Central Difference

$f^{\text{'}}(x)=\frac{f(x+h)-f(x-h)}{2h}+O(h^2)$

Forward Difference

$f^{\text{'}}(x)=\frac{f(x+h)-f(x)}{h}+O(h)$

Backward Difference

$f^{\text{'}}(x)=\frac{f(x)-f(x-h)}{h}+O(h)$

Second Derivative

The second order derivatives can be obtained by adding equations (2) and (3) (if properly expanded to include the fourth-derivative-term):

$f^{{\text{'}}^{\prime }}(x)=\frac{f(x+h)-2f(x)+f(x-h)}{h^2}+O(h^2)$