Compute all the directional derivatives of a trivariate polynomial function quickly

Question

Given a trivariate polynomial $A\in\mathbb{R}[x,y,z]$, a direction $\vec v\in\mathbb{R}^3$ and a point $p\in \mathbb{R}^3$, what is the fastest way to compute the directional deriviatives $\nabla_{\vec v}A(p), \nabla_{\vec v}\nabla_{\vec v}A(p),\dotsc$?

Answer 1

Since polynomials are smooth, you could use the simple dot product representation $$\nabla_\vec{v} A(p)= \langle \text{grad}_p(A), \vec{v} \rangle. $$