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# Numerical differentiation

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### Numerical differentiation

1. 1. NUMERICAL DIFFERENTIATION<br />
2. 2. <ul><li>We like to estimate the value of f '(x) for a given function f(x).
3. 3. The derivative represents the rate of change of a dependent variable with respect to an independent variable.
4. 4. The difference approximation is
5. 5. If x is allowed to approach zero, the difference becomes a derivative:</li></li></ul><li>f (Xi + ∆X)<br />∆y<br />f(Xi) <br />X<br />∆Xi<br />Xi + ∆X<br />∆x<br />
6. 6. Y<br />f (Xi + ∆X)<br />∆y<br />f(Xi) <br />∆Xi<br />Xi + ∆X<br />X<br />∆x<br />
7. 7. Y<br />f’(Xi)<br />Xi<br />X<br />
8. 8. <ul><li>The Taylor series expansion of f(x) about xi is
9. 9. From this:
10. 10. This formula is called the first forward divided difference formula and the error is of order O(h).</li></li></ul><li><ul><li>Or equivalently, the Taylor series expansion of f(x) about xi can be written as
11. 11. From this:
12. 12. This formula is called the first backward divided difference formula and the error is of order O(h).</li></li></ul><li><ul><li>A third way to approximate the first derivative is to subtract the backward from the forward Taylor series expansions:
13. 13. This yields to
14. 14. This formula is called the centered divided difference formula and the error is of order O(h2).</li></li></ul><li>FORWARD<br />True derivate<br />Y<br />Approximation<br />h<br />X<br />Xi<br />Xi+1<br />
15. 15. BACKWARD<br />True derivate<br />Y<br />Approximation<br />h<br />X<br />Xi-1<br />Xi<br />
16. 16. CENTERED<br />True derivate<br />Y<br />Approximation<br />2h<br />Xi+1<br />Xi-1<br />X<br />
17. 17. THANK YOU!<br />SOURCE: Dr Muhammad Al-Salamah, Industrial Engineering KFUPM<br />