Derive a formula for approximating the first derivative of an arbitrary function

Question

Derive a formula for approximating the first derivative of an arbitrary function at x = x0 x by interpolating at x0 + h and x0 - aaaah, for a > 0. What is the order of convergence for this formula? For what value(s) of a will it converge more quickl? (a is alpha)

Derive a formula for approximating the first derivative of an arbitrary function at x = xo by interpolating at xo + h and x0 - ah, for alpha > 0. What is the order of convergence for this formula? For what value(s) of a will it converge more quickly?

Explanation / Answer

using the definition of first derivative

f'(x) = h->0 [f(x+ah)-f(x)]/h

put x=x0-ah

f('x) = h->0 f(x0)-f(x-ah)/h

this coverges whenever a->infinity

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