Total Derivative - Application of The Total Differential To Error Estimation

Application of The Total Differential To Error Estimation

In measurement, the total differential is used in estimating the error Δf of a function f based on the errors Δx, Δy, ... of the parameters x, y, .... Assuming that the interval is short enough for the change to be approximately linear:

Δf(x) = f'(x) × Δx

and that all variables are independent, then for all variables,

This is because the derivative fx with respect to the particular parameter x gives the sensitivity of the function f to a change in x, in particular the error Δx. As they are assumed to be independent, the analysis describes the worst-case scenario. The absolute values of the component errors are used, because after simple computation, the derivative may have a negative sign. From this principle the error rules of summation, multiplication etc. are derived, e.g.:

Let f(a, b) = a × b;
Δf = faΔa + fbΔb; evaluating the derivatives
Δf = bΔa + aΔb; dividing by f, which is a × b
Δf/f = Δa/a + Δb/b

That is to say, in multiplication, the total relative error is the sum of the relative errors of the parameters.

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