Using points on either side of a point *x*_{0} results in the
best approximations of the derivative. Unfortunately, often in engineering,
especially in real-time applications, we must use only past data to estimate
the derivative at a point. For example, it may be necessary to use the current
rate-of-change of an incoming signal to adjust a system, but only past points
are available. Formula which use a technique similar to that in 13.1 and use only points ≤ *x*_{0} to
approximate the derivative at *x*_{0} are termed *backward
divided-difference formula*.

There are corresponding formulae using points greater than or equal to *x*_{0}, but
the derivation of these are left as an exercise to the reader.

# Background

Useful background for this topic includes:

# References

- Bradie, Section 6.2, Numerical Differentiation, Part II, p.438.
- Mathews, Section 6.2, Numerical Differentiation Formulas, p.343.
- Chapra, Section 23.1, High-accuracy Differentiation Formaulas, p.634.

# Interactive Maplet

To generate a backward divided-difference formula, keep the points to the left of *x*, for example, `f(x - 3*h)` to `f(x)`.

Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.