Chain rule
History

The chain rule seems to have first been used by Gottfried Wilhelm Leibniz. He used it to calculate the derivative of a + b z + c z 2 {\displaystyle {\sqrt {a+bz+cz^{2}}}} as the composite of the square root function and the function a + b z + c z 2 {\displaystyle a+bz+cz^{2}\!}. He first mentioned it in a 1676 memoir (with a sign error in the calculation). The common notation of the chain rule is due to Leibniz. Guillaume de l'Hôpital used the chain rule implicitly in his Analyse des infiniment petits. The chain rule does not appear in any of Leonhard Euler 's analysis books, even though they were written over a hundred years after Leibniz's discovery.. It is believed that the first "modern" version of the chain rule appears in Lagrange’s 1797 Théorie des fonctions analytiques; it also appears in Cauchy’s 1823 Résumé des Leçons données a L’École Royale Polytechnique sur Le Calcul Infinitesimal.
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