Calculus How To

Backward Differencing

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Numerical Differentiation >

graph backwards difference

You have to use backwards differences at the right endpoint of a function (at xi).

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:
backwards differencing

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

Example question: Approximate the derivative of f(x) = x2 + 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) = 32 + 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) = 22 + 2(2) = 8

Step 3: Insert your values into the formula and solve:
backwards differencing example


Andasari, V. (2020). Numerical Differentiation. Retrieved September 21, 2020 from:
Kutz, J. (2013). Data-Driven Modeling & Scientific Computation. Methods for Complex Systems & Big Data. OUP Oxford.

Stephanie Glen. "Backward Differencing" From Calculus for the rest of us!

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