Page:Annales de mathématiques pures et appliquées, 1829-1830, Tome 20.djvu/237

 Les corrections sont expliquées en page de discussion

${\frac {\operatorname {d} X}{X}}={\frac {\alpha }{x}},\qquad {\frac {\operatorname {d} X'}{X'}}={\frac {\alpha '}{x}},$

${\frac {\operatorname {d} .XX'}{XX'}}={\frac {\alpha +\alpha '}{x}},$

donc, en comparant,

${\frac {\operatorname {d} .XX'}{XX'}}={\frac {\operatorname {d} X}{X}}+{\frac {\operatorname {d} X'}{X'}},\qquad$(8)

ou encore

$\operatorname {d} .XX'=X'\operatorname {d} X+X\operatorname {d} X'.\qquad$(9)

Soit toujours $X$ une fonction monome, et soit $X'$ une fonction polynome ; telle qu’on ait

$X'=X'_{1}+X'_{2}+X'_{3}+\ldots \,;$

$X'_{1},X'_{2},X'_{3},\ldots ,$ étant des monomes, on aura

$XX'=XX'_{1}+XX'_{2}+XX'_{3}+\ldots \,;$

de là on conclura (2)

$\operatorname {d} X'=\operatorname {d} X'_{1}+\operatorname {d} X'_{2}+\operatorname {d} X'_{3}+\ldots ,$

$\operatorname {d} .XX'=\operatorname {d} .XX'_{1}+\operatorname {d} .XX'_{2}+\operatorname {d} .XX'_{3}+\ldots \,;$

mais on a (9)

${\begin{aligned}&\operatorname {d} .XX'_{1}=X'_{1}\operatorname {d} X+X\operatorname {d} X'_{1},\\&\operatorname {d} .XX'_{2}=X'_{2}\operatorname {d} X+X\operatorname {d} X'_{2},\\&\operatorname {d} .XX'_{3}=X'_{3}\operatorname {d} X+X\operatorname {d} X'_{3},\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\end{aligned}}$