## Topic 12.4: Higher-Order Derivatives (HOWTO)

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# Problem

Approximate a higher derivative of a univariate function f(x) at a point x0. We will assume that we are given a sequence of points (xi, f(xi)) around the point of interest (either before or around). We will not look at iteration because the process of Richardson extrapolation converges significantly faster.

# Assumptions

We need to assume the function has an nth derivative if we are to bound the error on our approximation.

# Tools

We will use interpolation.

# Process

If we are to evaluate the second derivative at the point (xi, f(xi)) and have access to the two surrounding points, (xi − 1, f(xi − 1)) and (xi + 1, f(xi + 1)), then we may find the interpolating polynomial, differentiate it twice, and evaluate that derivative at xi:

This is simply another form of the formula

where h is the distance between the points, that is, h = xi - xi − 1.

If we have access to two points on either side of xi, we can calculate

where h = xi - xi − 1.

This is another form of the formula:

We could perform the same operations with higher derivatives, however, it should be noted that to calculate the nth derivative, we require at least n + 1 points.