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.
Next by cutting off the Taylor series after the fourth term and evaluating it at h and -h yields Equations (2) and (3).
Then by subtracting Equation (2) by Equation (3) yields.
The second order derivatives can be obtained by adding equations (2) and (3) (if properly expanded to include the fourth-derivative-term):