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

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 $u,$ v,  w,  $\dots . \$