**Backward differencing**is a way to estimate a derivative with a range of x-values. The algorithm “moves” the points closer and closer together until they resemble a tangent line.

**Other types of differencing are forward differencing, and central differencing.** When h (the distance between the two points) is greater than zero (i.e. h > 0), you have forward differencing. When h < 0, you have backward differencing. An average of the two methods (using three points) is called central differencing. For a given number of points, **central differencing is more accurate**. However, you can’t use central differencing at the left and right endpoints because there are no points further to the left or right with which to take an average. These methods come in useful when you have no data from the past or in the future beyond certain points. At these points you have to use forward differencing (left endpoint) and backwards differencing (right endpoint).

## Backward Differencing Formula

The formula for backward differencing is:

If you’ve calculated slopes before, you’ll find the calculation very familiar.

**Example question**: Approximate the derivative of f(x) = x^{2} + 2x at x = 3 using backwards differencing with a step size of 1.

Step 1: **Calculate the function value at the given point.** For this example, that’s x = 3:

f(3) = 3^{2} + 2(3) = 15

Step 2: **Find the function value at one step behind.** We’re given that the step size is 1 in the example problem, so we’re calculating the value at x = 2:

f(2) = 2^{2} + 2(2) = 8

Step 3: **Insert your values into the formula and solve:**

## References

Andasari, V. (2020). Numerical Differentiation. Retrieved September 21, 2020 from: http://people.bu.edu/andasari/courses/Fall2015/LectureNotes/Lecture7_24Sept2015.pdf

Kutz, J. (2013). Data-Driven Modeling & Scientific Computation. Methods for Complex Systems & Big Data. OUP Oxford.

**CITE THIS AS:**

**Stephanie Glen**. "Backward Differencing" From

**CalculusHowTo.com**: Calculus for the rest of us! https://www.calculushowto.com/backward-differencing/

**Need help with a homework or test question? **With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!