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Question about basis of $\text{Der}_{k}(k[X])$

Let $k[X] = k[x_1,\ldots, x_n]$ be the polynomial ring over a field of characteristic zero. Assume that $(D_1,\ldots, D_n)$ is a $k[X]$-basis of $\text{Der}_k(k[X])$. Suppose that the vector space $\...
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