Advertisement

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

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 . \$$

## Copyright information

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

1. 1.Sun CityUSA