I am just posting my comment as one answer. Using the local flatness criterion, the ring $R/\langle \partial_1(f),\dots,\partial_n(f)\rangle$ is Cohen-Macaulay if and only if it is flat as a module over $\mathbb{C}\{x_n\}$, i.e., if and only if multiplication by $x_n$ is injective on the ring. In the answer to a previous question, I wrote an example where multiplication by $x_n$ is not injective on the ring, i.e., I wrote an example where the image of $x_n$ is a zero divisor in the ring. Here is the link to the previous question: Deformation of isolated singularities and non zero divisors