1. 1 Chapter 6 NUMERICAL DIFFERENTIATION

2. 2 When we have to differentiate a function given by a set of tabulated values or when a complicated function is involved, numerical differentiation is used. The principle behind numerical differentiation is “we find a suitable interpolation polynomial passing through the given data points and then the polynomial is differentiated analytically as required”.

3. 3 DERIVATIVES FOR EQUALLY SPACED DATA 1. Newton Forward Differentiation

4. 4 Newton’s Forward Differentiation Formulae = =

5. 5 Differentiating

6. 6 Newton’s Backward Differentiation Formulae = =

7. 7 Particular case In particular, when x = x 0, we get p = 0. = =

8. 8 Example Find the first two derivatives at x = 1.1 from the following data: x:11.21.41.61.82.0 y:0.00000.12800.54401.29602.43204.0000

9. 9 Solution The difference table is:

10. 10 p = = 0.5 and h = 0.2 Applying Newton’s forward differentiation formula = 0.630 = 6.60

11. 11 Example The following table gives the displacement in metres at different times. Find the velocities and accelerations at t = 1.8 s. t:00.51.01.52.0 s:08.7530.0071.25140.00

12. 12 Solution h = 0.5, p = = - 0.4 The difference table is:

13. 13 Applying Newton’s backward differentiation formula

14. 14 UNEQUAL INTERVALS For unequally spaced data, Lagrange’s interpolation formula may be differentiated. Example Find the first derivative at x = 2 for the function given by the data x11.52.03.0 y00.40570.693151.09861

15. 15 Solution Lagrange interpolation polynomial is: y x = x 0 + x 0.4057 + x 0.69315 + x 1.09861 Differentiating and substituting for x as 2.