Calculus How To

Numerical Differentiation

Derivatives >

Many known functions have exact derivatives. For example, the derivative of the natural logarithm, ln(x), is 1/x. Other functions involving discrete data points don’t have known derivatives, so they must be approximated using numerical differentiation. The technique is also used when analytic differentiation results in an overly complicated and cumbersome expression (Bhat & Chakraverty, 2004).

Interpolation as a Numerical Differentiation Method

One of the easiest ways to approximate a derivative for a set of discrete points is to create an interpolation Function, which gives an estimated continuous function for the data. Once you have the estimated function (for example a polynomial function), you can then use known derivatives.

If the function is given as a table of values, a similar approach is to differentiate the Lagrange interpolation formula to get (Ralston & Rabinowitz, 2001):
numerical differentiation

Numerical Differentiation using Differences

Differences are a set of tools for estimating the derivative using a set range of x-values. The basic idea is that the algorithms “move” the points so that they get closer and closer together, to look like a tangent line. Exactly how the points are moved (for example, forwards or backwards) gives rise to three common algorithms: backward differencing, forward differencing, and central differencing.

  • Forward differencing (a one-sided differencing algorithm) is based on values of the function at points x and x + h.
  • Backward differencing (also one-sided)is based on values at x and x – h.
  • Central (or centered) differencing is based on function values at f(x – h) and f(x + h).

While all three formulas can approximate a derivative at point x, the central difference is the most accurate (Lehigh, 2020).

Let’s say you have a table with the following values, and you want to approximate the derivative at x = 0.5 using the central difference.
central difference examples

Using the formula from above, you would get:
F′(x) = (0.71 – 0.25) / 0.25 = 0.92.

References

Bhat, R. & Chakraverty, S. (2004). Numerical Analysis in Engineering. Alpha Science International.
Hoffman, J. & Frankel, S. (2001). Numerical Methods for Engineers and Scientists, Second Edition. Taylor & Francis.
Lehigh University (2020). Numerical Differentiation. Retrieved September 6, 2020 from: https://www.lehigh.edu/~ineng2/clipper/notes/NumDif.htm
Levy, D. Numerical Differentiation. Retrieved September 6, 2020 from: http://www2.math.umd.edu/~dlevy/classes/amsc466/lecture-notes/differentiation-chap.pdf
Ralston, A. & Rabinowitz, P. (2001). A First Course in Numerical Analysis. Dover Publications.
MIT: Finite Difference Approximations.

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