Es sei ${\displaystyle {}K}$ ein Körper und seien ${\displaystyle {}F_{1},\ldots ,F_{m}\in K[X_{1},\ldots ,X_{n}]}$ und ${\displaystyle {}G_{1},\ldots ,G_{k}\in K[Y_{1},\ldots ,Y_{m}]}$ Polynome. Wir setzen

${\displaystyle {}H_{i}=G_{i}(F_{1},\ldots ,F_{m})\,.}$

Zeige, dass die formalen partiellen Ableitungen die „formale Kettenregel“

${\displaystyle {\begin{pmatrix}{\frac {\partial H_{1}}{\partial X_{1}}}&\ldots &{\frac {\partial H_{1}}{\partial X_{n}}}\\\vdots &\ddots &\vdots \\{\frac {\partial H_{k}}{\partial X_{1}}}&\ldots &{\frac {\partial H_{k}}{\partial X_{n}}}\end{pmatrix}}={\begin{pmatrix}{\frac {\partial G_{1}}{\partial Y_{1}}}&\ldots &{\frac {\partial G_{1}}{\partial Y_{m}}}\\\vdots &\ddots &\vdots \\{\frac {\partial G_{k}}{\partial Y_{1}}}&\ldots &{\frac {\partial G_{k}}{\partial Y_{m}}}\end{pmatrix}}{\left({\frac {F_{j}}{Y_{j}}}\right)}\circ {\begin{pmatrix}{\frac {\partial F_{1}}{\partial X_{1}}}&\ldots &{\frac {\partial F_{1}}{\partial X_{n}}}\\\vdots &\ddots &\vdots \\{\frac {\partial F_{m}}{\partial X_{1}}}&\ldots &{\frac {\partial F_{m}}{\partial X_{n}}}\end{pmatrix}}\,}$

${\displaystyle {}{\frac {F_{j}}{Y_{j}}}}$

${\displaystyle {}Y_{j}}$

${\displaystyle {}F_{j}}$