Derivatives Using Newton's Backward Difference Formula

At x = xn-1, we get s = – 1. Hence, we obtain the approximation to the second derivative f ″(xn-1) as


We use the formulas (1.4), (1.5), (1.7) and (1.8) when the entire data is to be used.

Note :  We use the backward difference formulas for derivatives, when we need the values of the derivatives near the end of table of values.


Example :  Using the operator relation, derive approximations to the derivatives f′(xn), f″(xn) in terms of the backward differences.

Solution:   From the operator relation E = ehd, where D = d/dx, we obtain