SKEDSOFT
This method is used to find the derivatives of higher order divided difference.
Derivatives using Divided Difference Formula :
The divided difference interpolation polynomial fitting the data (xi, f(xi)), i = 0, 1, 2,…, n is given by
.........................1.1
Differentiating with respect to x, we get
.......................1.2
If the derivative f ′(x) is required at any particular point x = x*, then we substitute x = x* in (1.3). If the data is equispaced, then the formula is simplified.
Differentiating (1.2) again, we obtain
............................1.3
If the second derivative f ″(x) is required at any point x = x*, then we substitute x = x* in (1.3). Again, if the data is equispaced, then the formula is simplified.
However, we can also determine the Newton’s divided differences interpolation polynomial and differentiate it to obtain f ′(x) and f ″(x).
Example : Find the first and second derivatives at x = 1.6, for the function represented by the following tabular data:
Solution : The data is not equispaced. We use the divided difference formulas to find the derivatives. We have the following difference table:
Divided Differences
Substituting x = 1.6 in the formula
