## Numerical Differentiation

**Numerical Differentiation:
**

The general method for deriving the numerical differentiation formulae is to differentiate the interpolating polynomial. Here we illustrate the derivation with Newton’s forward difference formula only. The method of derivation being the same with regard to the other formulae. Consider Newton’ forward difference formula:

This formula is for nontabular values of x. For tabular values of x, by setting x = x_{0} we obtain u = 0 from above equation. Hence equation gives;

Again for second order derivative, differentiating above equations further, we obtain

from which, we get

Similarly other higher order derivatives may be obtained by successive differentiation. In a similar way, different formulae can be derived by starting with other interpolation formulae. Thus Newton’s backward interpolation formula gives;

**Numerical Integration:
**

The process of computing, where y = f (x) is given by a set of tabulated values [x_{i}, y_{i}] where i = 0,1,2, ...,n, a = x_{0} and b = x_{n} is called Numerical Integration. Since y = f (x) is a single variable function, the process in general is known as quadrature.