Let $U \subseteq \mathbb{C}[x_1,\dots, x_n]_d$ be a linear subspace of the vector space of homogeneous degree-$d$ polynomials (including zero). I would like a proof or counterexample of the claim that for a general linear partial derivative $\partial=\alpha_1 \frac{\partial}{\partial{x_1}}+\dots + \alpha_n \frac{\partial}{\partial{x_n}}$, it holds that the linear subspace $\partial U \subseteq \mathbb{C}[x_1,\dots, x_n]_{d-1}$ obtained by applying $\partial$ to every element of $U$ satisfies the following dimension bound: $$ \binom{n+d-1}{d}\dim(\partial U) \geq \binom{n+d-2}{d-1} \dim(U) $$