CHALLENGE -- Defining an ORIGINAL second derivative function?

Questions
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1

I began with this definition of the derivative

Limit Definition of Derivative,

which resulted in: Limit[ #, h → 0] & /@ { (f[x + h, y] - f[x, y])/ h.

But, I want to calculate the derivative at a point with the definition

Derivative at a Point.

One aim is to increase the accuracy of the calculation by using a set of set of xi (a numerical grid), where the spacing (the distance from one grid point to the next) is ?x. For example, ?x = 0.001, a grid of 10,000 points (i = 1, . . . , 10000) would span a spatial range of 10 units in x.

I’m thinking one of these forms would be ideal for xi …

x<sub>i</sub> = i?x
x<sub>i</sub> = (i ? i<sub>m</sub>) ?x
x<sub>i</sub> = i?x ? x<sub>m</sub>,

but I’m not sure which is best.

Ultimately, I want to take a function f of two variables, for example

f[x_, n_] := - Sum [Sin[j x] /j, {j, 1, n}],

at the grid points xi, and run through the grid, applying the derivative at a point equation (involving two neighbors for each derivative point). Then, after having the first derivative on the entire grid, calculate the second derivative on the grid, involving two first-derivative neighbors. Finally, I want to see the beautiful results on a plot.

I’m sure there will be a do loop, couple arrays, and ListPlot function involved. I’m just not sure how to do all this, as I’ve just started with Mathematica.

We have any good solutions?

2

You do know that Mathematica can compute derivatives:

f’[x]=D[f[x],x] and f’'[x]=D[f[x],{x,2}]

3

UPDATE: I do not want to use the Derivative or D functions. –

4

UPDATE: I do not want to use the Derivative or D functions. –

You must not, you can use Cauchy’s integral to differentiate, see Complex variable methods in Numerical Differentiation, you can use the ND implementation in the Mathematica package NumericalCalculus` …