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

  • Dennis M. CatesEmail author
Chapter

Abstract

Let \(s=F(u, v, w, \dots )\) be 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 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 \( u, \) vw\( \dots . \ \)

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Sun CityUSA

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