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The generalized chain rule of differentiation with historical notes. (English) Zbl 1004.26003

There is a not insubstantial literature on the problem of computing the \(n\text{-th}\) derivative of \(f\circ z\) when \(z:[c,d]\to[a,b]\) and \(f:[a,b]\to \mathbb C\) are each \(n\text{-times}\) differentiable. In fact, the author of this paper is preparing a 150-item bibliography. The result most often cited is attributed to Faà de Bruno, in Italian publications of 1855 and 1857. But the author’s sleuthing has uncovered the fact that it was published 5 years earlier in French papers of Theodore Anne. A less well-known but significantly more transparent formula was published even earlier, in 1845 (Crelle’s Journal) and again in 1871 [Clebsch Ann. IV, 85-87 (1871; JFM 03.0115.02)] by R. Hoppe. It reads \[ D^n_x f(x)= \sum^n_{k=0} D^k_z f(x) \frac{1}{k!} \sum^k_{j=0} (-1)^{k-j}) \binom kj z^{k-j} D^n_x z^j. \tag{1} \] The author offers a discovery path to and a proof of (1). For the special case that \(f(z):= z^\alpha\), \(\alpha\) real, (1) yields \[ z^{-\alpha} D^n_x z^\alpha = \sum^n_{j=0} z^{-j} D^n_x z^j \prod_{\substack{ k=0\\k\not= j}}^n \frac{\alpha-k}{j-k} \tag{2} \] which the author compares to Lagrange’s interpolation formula \[ f(\alpha)=\sum^n_{j=0} f(x_j) \prod _{\substack{ k=0\\ k\not= j}}^n \frac{\alpha-x_k}{x_j -x_k} \tag{3} \] expressing the \(n\)th-degree polynomial \(f(\alpha)\) in terms of its values at \(n\) distinct points \(x_j\): Formula (2) is formula (3) applied to the function \(f(\alpha):=z^{-\alpha} D^n_x z^\alpha\) with \(x_j:=j\), and it really is an interpolation formula because it recovers the derivatives of an arbitrary real power from knowledge of the derivatives of non-negative integer powers. Finally, the case \(\alpha=-1\) is especially useful and simple. It reads \[ D^n_x \left(\frac{1}{z}\right) =\sum^n_{j=0} (-1)^j \binom {n+1}{j+1} z^{-j-1} D^n_x z^j. \]

MSC:

26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26-03 History of real functions
01A55 History of mathematics in the 19th century

Citations:

JFM 03.0115.02
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