Suppose that we are given an array of values , where i=0,1,2,...,N. We assume that there exists a continuous function underlying the N+1 values. Further, we assume that the underlying function has as many derivatives defined as is necessary for our purposes below. We also assume that there are N+1 values of a variable x, . For simplicity we will take the to be uniformly distributed in values with and the separation of and independent of i, so for all i. Having the non-uniformly distributed would complicate the considerations which follow.
The problem is then to find approximate expressions for the first derivative using only the provided data .
Given our assumption that there is an underlying continuous function with as many derivatives as needed, we make use of the familiar Taylor series expansion formula, which expresses the function f as an infinite series expansion about a given point of interest:
Let's evaluate f at using the Taylor series. Simplifying slightly we find:
Now if we let be , then corresponds to and the Taylor series expansion becomes that for . Note that if we subtract from Eq.(2) for the i+1 case, we have
Dividing by dx and solving for the first derivative at we obtain the forward difference approximation for the first derivative:
Note that the error in this approximation is of order in the separation between points. Such an dependence is said to indicate that the forward difference approximation is only "first order accurate". As we will see below, one can do better than this.
Next, we give the result of performing the calculation as above except using and . Then, taking the difference between and fi from the Taylor series expression and simplifying a bit, we find the backward difference approximation for the first derivative:
As is clear from inspecting this result, the backward difference approximation is also only first order accurate in dx.
Before we proceed to find more accurate approximations, let's pause to make a comment on these two formulae. From the perspective of order of accuracy, the forward and backward difference formulae represent the simplest and least accurate approximations that one can imagine. In practice these formulae are not used except in situations where a quick estimate of the derivative is needed. In most applications in computational physics, the resulting low accuracy that these formulae present make them just not good enough for serious simulation. While the two formulae are not widely used and thus will not get much attention from us in future discussions, here is a good place to point out another aspect of them which may need attention when we get some higher order accurate approximations: their one-sidedness. To illustrate the point, we can simply note that if we used the forward difference formula we would run into trouble when we try to use it on the last point i=N. There is no point and function value at i=N+1. So we can not compute the derivative at the rightmost point. However, we could get a first order accurate estimate there if we used the backward difference approximation just for that last point, since it uses values of the function at i and i-1. A similar remark can be made about the leftmost point . At i=0 we can not use the backward formula but could use the forward formula. Hence, to the same order of accuracy in dx if we utilize both formulae we can find the derivative at all points .
In addition, let's note that the forward and backward difference approximations would be exact if the underlying function f(x) happened to be a linear function f(x) = a + b x. However, in applications one does not know the underlying function and thus can not determine, in advance, whether these forward and backward formulae are the only ones needed. In most situation, they certainly do not suffice to capture enough about the rate of change of the data to be very useful.