Page 126 - Excel for Scientists and Engineers: Numerical Methods

CHAPTER 6 DIFFERENTIATION 103

There are other equations for numerical differentiation that use three or more
points instead of two points to calculate the derivative. Since these equations
usually require equal intervals between points, they are of less generality. Again,
their main advantage is that they minimize the effect of "noise." Table 6-1 lists
equations for the first, second and third derivatives, for data from a table at
equally spaced interval h.
These difference formulas can be derived from Taylor series. Recall from
Chapter 4 that the first-order approximation is
F(x + h) N F(x) + hF'(x) (6-5)
or, in the notation used in Table 6-1
YI+1 = YI + hY', (6-6)

which, upon rearranging, becomes

admittedly, an obvious result.
The second derivative can be written as

When each of the y' terms is expanded according to the preceding expression
for y', the expression for the second derivative becomes

or

(6-10)

The same result can be obtained from the second-order Taylor series
expansion
h2
F(x + h) is F(x) + hF'(x) + -FF"(x) (6-1 1)
2!
which is written in Table 6-1 as
I h2 ,,
Yl+1 = Y, + hY1 +ZYI (6-12)

by substituting the backward-difference formula for F from Table 6-1.
Expressions for higher derivatives or for derivatives using more terms can be
obtained in a similar fashion.