Corollary 3

Tag 06CB, permanent. Cite it as [Bourbaki, Tag 06CB].

Algebra, chapter IV, § 4. Formal power series

— Let $f \in \mathbf{A}[[X_1, ..., X_s]]$ and $g_i \in \mathbf{A}[[Y_1, ..., Y_q]]$ for $1 \leq i \leq p$ . Suppose that for $1 \leq i \leq p$ the constant term of $g_i$ is zero, and put $h = f(g_1, ..., g_p)$ . Then for $1 \leq j \leq q$ we have

\frac{\partial h}{\partial Y_j} = \sum_{i=1}^p D_i f(g_1, ..., g_p) \cdot \frac{\partial g_i}{\partial Y_j} .

This is the special case $E = \mathbf{A}[[Y_1, ..., Y_s]]$ , $x_i = q_i$ and $A = \partial / \partial Y_j$ of Cor. 1.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Proposition 14, § 4

Cites

Cor. 1