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The problem of finding where a function /(*) that is tabulated at equal intervals h has a given derivative (i.e. given /'(*) = fj — /'(*o + ph), to find p) arises very frequently in numerical computation, especially in the location of a maximum or minimum point when /'(*o + ph) = 0. The usual methods for finding where the derivative of a function has a certain value are either (1) to calculate the derivatives at the tabular points and then perform inverse interpolation, (2) to estimate where the derivative has the desired value and then subtabulate the first derivative in which connection a table by Salzer1 will be found helpful, or (3) to reverse the series in p, obtained by differentiating any one of the direct interpolation formulas for /(*o + ph). The drawback in (1) is that it involves two separate steps, and the drawback in (2) is that it requires much more work than is necessary, together with a large amount of continual personal inspection. Thus method (3) which gives p in terms of the tabular entries /(*o + mh) seems to be the most convenient, and it is recommended by Gibb,2 in the form of reversion of a series involving differences. This present note gives the coefficients of the reversed series for p directly in terms of the tabular entries, so that no differences are necessary. However, if differences are already available, one might use them conveniently in the formulas given at the end which are obtained by reversing the series for the derivative of the Newton-Stirling formula using central differences 502m and mean-central differences p502m+1, viz.,

Formulas for finding the argument for which a function has a given derivative