Let $V$ be the space of all homogenous polynomials over $\mathbb{C}$ in $n$ variables of degree $d$. 
Let $l,k$ be  two integers and $f\in V$. 
Let $\partial^{=k}(f)$ be the space of all partial derivatives of $f$ of degree exactly $k$.
I want to understand how the space of partial derivatives looks like. For example if we will allow to to multiply the partial derivatives by polynomial of degree up-to $l$ what the dimention of the space will we get? 
Let us define the space $$L_{k,l}(f)=span \left[ a(x)p(x): deg (a(x))=l, p(x)\in \partial^k(f)\right].$$

I wonder if it is possible to calculate the dimension of $L_{k,l}(f)$ for a typical $f$.