Numerical Differentiation

1.Numerical differentiation is the process of finding the numerical value of a derivative of a given function at a given point.

        2.The simplest way to compute a function’s derivatives numerically is to use finite difference approximations. Suppose we are interested in computing the first and second derivatives of a smooth function f: R→R

Then The definition of a derivative,

suggests a natural approximation. Take a small number h, (more on how small latter) and

....................1.1

        3. Approximation to the derivatives can be obtained numerically using the following two approaches.

           (i) Methods based on finite differences for equispaced data.
           (ii) Methods based on divided differences or Lagrange interpolation for non-uniform data.

        4.This is the easiest and most intuitive finite difference formula and it is called the forward  difference.

        5. The forward difference is the most widely used way to compute numerical derivatives but often it is not the best choice.

        6. In order to compare to alternative approximations we need to derive an error bound for the forward difference.

        7.  This can be done by taking a Taylor expansion of f,