Z_1,\dots,Z_n\rangle \otimes {\mathbb{C}}\langle Z_1,\dots,Z_n\rangle$$ that satisfies $\partial_j Z_i = \delta_{i,j} 1 \otimes 1$ for $i=1,\dots,n$.

Recall that a linear map $\delta: {\mathcal{A}}\rightarrow {\mathcal{M}}$, which is defined on a complex unital algebra ${\mathcal{A}}$ and takes its values in a ${\mathcal{A}}$-bimodule ${\mathcal{M}}$, is called a derivation if it satisfies the Leibniz rule $$\delta(a_1 a_2) = \delta(a_1) \cdot a_2 + a_1 \cdot \delta(a_2) \qquad\text{for all $a_1,a_2\in{\mathcal{A}}$}.$$

In particular, being a derivation means for $\partial_j$ that $$\label{derivation-def}
\partial_j (P_1 P_2) = (\partial_j P_1) (1 \otimes P_2) + (P_1 \otimes 1) (\partial_j P_2)$$ holds for all $P_1,P_2\in {\mathbb{C}}\langle Z_1,\dots,Z_n\rangle$. According to the ${\mathbb{C}}\langle Z_1,\dots,Z_n\rangle$-bimodule structure of ${\mathbb{C}}\langle Z_1,\dots,Z_n\rangle \otimes {\mathbb{C}}\langle Z_1,\dots,Z_n\rangle$, we will often abbreviate to $$\partial_j (P_1 P_2) = (\partial_j P_1) P_2 + P_1 (\partial_j P_2).$$

More explicitly, $\partial_j$ acts on monomials $P$ as $$\partial_j P = \sum_{P = P_1 Z_j P_2} P_1 \otimes P_2,$$ where the sum runs over all decompositions $P=P_1Z_jP_2$ of $P$ with monomials $P_1,P_2$.

We record here the easy but useful observation that each derivation $\delta: {\mathbb{C}}\langle Z_1,\dots,Z_n\rangle \rightarrow {\mathcal{M}}$ taking values in any ${\mathbb{C}}\langle Z_1,\dots,Z_n\rangle$-bimodule ${\mathcal{M}}$ is determined by its values $\delta(Z_1),\dots,\delta(Z_n)$, which can be expressed explicitly by $$\label{derivation-uniqueness}
\delta(P) = \sum^n_{j=1} \partial_j(P) \sharp \delta(Z_j) \qquad\text{for all $P\in{\mathbb{C}}\langle Z_1,\dots,Z_n\rangle$}.$$ 