Let $K$ be a field and $V$ be a linear space over $K$. A map $p\colon V \to K$ is homogeneous polynomial of degree $n$ if there exist the symmetric $n$-linear form $f\colon V^{\times n}\to K$ such that $p(x)=f(x,x,\ldots,x)$ for any $x\in V$.

The question: where (what books or articles?) can I read about the map $\delta_a p\colon V \to K$, $a\in V$, such that $\delta_a p(x)=nf(a,x,\ldots,x)$ for any $x\in V$? As far as I understand, the map is the derivative of $p$ along $a$, is it? So I probably need some papers related to differential geometry?

Update 1: $\delta_a p(x)=f(a,x,\ldots,x)$ fixed, the correct one is $\delta_a p(x)=nf(a,x,\ldots,x)$ (sorry for the erratum)