USE OF PARTIAL DERIVATIVES IN THE DIFFERENTIATION OF COMPOSED FUNCTIONS. DIFFERENTIALS OF IMPLICIT FUNCTIONS

Dennis M. Cates ()

Chapter Chapter 9 in Cauchy's Calcul Infinitésimal, 2019, pp 43-47 from Springer

Abstract: Abstract Let $$s=F(u, v, w, \dots )$$ beImplicit function Partial derivative any function of the variable quantities $$ u, v, w, \dots $$ that we suppose to be themselves functions of the independent variables $$ x, y, z, \dots . \ $$ s will be a composed functionComposed function of these latter variables; and, if we designate by $$ \varDelta x, $$ $$ \varDelta y, $$ $$ \varDelta z, $$ $$ \dots $$ the arbitrary simultaneous increments attributed to $$ x, y, z, \dots , $$ the corresponding increments $$ \varDelta u, \varDelta v, \varDelta w, \dots , \varDelta s $$ of the functions $$ u, v, w, \dots , s $$ will be related among themselves by the formula $$\begin{aligned} \varDelta s = F(u+\varDelta u, v+\varDelta v, w+\varDelta w, \dots )-F(u, v, w, \dots ). \end{aligned}$$ Moreover, let $$\begin{aligned} \varPhi (u, v, w, \dots ), \ \ \ X(u, v, w, \dots ), \ \ \ \varPsi (u, v, w, \dots ), \ \ \ \dots \end{aligned}$$ be the partial derivatives of the function $$F(u, v, w, \dots )$$ taken successively with respect to $$ u, $$ v, w, $$ \dots . \ $$

Date: 2019
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DOI: 10.1007/978-3-030-11036-9_9

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