Es sei ${\displaystyle {}R}$ ein kommutativer Ring, ${\displaystyle {}A}$ eine kommutative ${\displaystyle {}R}$-Algebra, ${\displaystyle {}M}$ ein ${\displaystyle {}R}$-Modul und ${\displaystyle {}D\colon A\rightarrow M}$ eine ${\displaystyle {}R}$-Derivation. Es sei ${\displaystyle {}x_{1}^{n_{1}}\cdots x_{r}^{n_{r}}\in A}$. Zeige

${\displaystyle {}D{\left(x_{1}^{n_{1}}\cdots x_{r}^{n_{r}}\right)}=n_{1}x_{1}^{n_{1}-1}x_{2}^{n_{2}}\cdots x_{r-1}^{n_{r-1}}x_{r}^{n_{r}}D{\left(x_{1}\right)}+\cdots +n_{r}x_{1}^{n_{1}}\cdots x_{r-1}^{n_{r-1}}x_{r}^{n_{r}-1}D{\left(x_{r}\right)}\,}$