The document discusses numerical methods for estimating the derivative of a function f(x) at a point x=xi. It introduces three approaches: 1) the forward difference approximation calculates the slope between xi and xi+h, 2) the backward difference approximation calculates the slope between xi-h and xi, and 3) the centered difference approximation calculates the average of the forward and backward slopes. Each method has an error term that approaches zero as h approaches zero.

1. NUMERICAL DIFFERENTIATION

3. The derivative represents the rate of change of a dependent variable with respect to an independent variable.

4. The difference approximation is

6. Y f (Xi + ∆X) ∆y f(Xi) ∆Xi Xi + ∆X X ∆x

7. Y f’(Xi) Xi X

9. From this:

11. From this:

13. This yields to

15. BACKWARD True derivate Y Approximation h X Xi-1 Xi

16. CENTERED True derivate Y Approximation 2h Xi+1 Xi-1 X

17. THANK YOU! SOURCE: Dr Muhammad Al-Salamah, Industrial Engineering KFUPM