247
DES FONCTIONS.
![{\displaystyle \left.{\begin{aligned}&{\frac {\operatorname {d} U}{\operatorname {d} a}}{\frac {\operatorname {d} V}{\operatorname {d} b}}={\frac {\operatorname {d} U}{\operatorname {d} b}}{\frac {\operatorname {d} V}{\operatorname {d} a}}\\\\&{\frac {\operatorname {d} U}{\operatorname {d} a'}}{\frac {\operatorname {d} V}{\operatorname {d} b'}}={\frac {\operatorname {d} U}{\operatorname {d} b'}}{\frac {\operatorname {d} V}{\operatorname {d} a'}}\end{aligned}}\right\}\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a54ad6f94e3e2ea1094a7924c6ea9cd822fd5734)
(5)
d’où on déduirait
![{\displaystyle \left.{\begin{aligned}&{\frac {\operatorname {d} .V{\frac {\operatorname {d} U}{\operatorname {d} a}}}{\operatorname {d} b}}={\frac {\operatorname {d} .V{\frac {\operatorname {d} U}{\operatorname {d} b}}}{\operatorname {d} a}}\\\\&{\frac {\operatorname {d} .V{\frac {\operatorname {d} U}{\operatorname {d} a'}}}{\operatorname {d} b'}}={\frac {\operatorname {d} .V{\frac {\operatorname {d} U}{\operatorname {d} b'}}}{\operatorname {d} a'}}\end{aligned}}\right\}\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f89994d8ed79e43e11a145de120dbd44c9542981)
(6)
et aussi
![{\displaystyle {\begin{aligned}&{\frac {\operatorname {d} .U{\frac {\operatorname {d} V}{\operatorname {d} a}}}{\operatorname {d} b}}={\frac {\operatorname {d} .U{\frac {\operatorname {d} V}{\operatorname {d} b}}}{\operatorname {d} a}},\\\\&{\frac {\operatorname {d} .U{\frac {\operatorname {d} V}{\operatorname {d} a'}}}{\operatorname {d} b'}}={\frac {\operatorname {d} .U{\frac {\operatorname {d} V}{\operatorname {d} b'}}}{\operatorname {d} a'}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f500a306fc45117d70c767c33a8fd1893d3a8746)
Les équations (4, 6) peuvent être utilement employées au développement de certaines fonctions. Soit, par exemple,
à développer suivant les puissances de
lorsque
est donnée par l’équation
![{\displaystyle x=\operatorname {f} (a+bz)\,;\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3df3275d6feb6af2fba3a992db356259992e5cf1)
(7)
désignant une fonction quelconque de
sans
ni
Nous déduirons d’abord de l’équation (7)
![{\displaystyle {\frac {\operatorname {d} x}{\operatorname {d} a}}=\left(1+b{\frac {\operatorname {d} z}{\operatorname {d} x}}{\frac {\operatorname {d} x}{\operatorname {d} a}}\right)\operatorname {f} '(a+bz)\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a0025733eb6091a1dc9c5fe3aabf350107fd2a)
désignant ici la fonction prime de
cela donnera