Let $X=V(F)\subset\mathbb{P}^{n}$ be a smooth irreducible hypersurface. Let us consider the morphism $$ \mathcal{G}:X\rightarrow \mathbb{P}^{N}, p\mapsto \left( \frac{\partial F}{\partial X_{0}}(x):\ldots:\frac{\partial F}{\partial X_{N}}(x) \right). $$ Notice that we may understand it as the Gauss map $$ X\rightarrow (\mathbb{P}^{N})^{*}, p\mapsto T_{p}X. $$ Given $p\in X$, I was wondering what is known about de differential map $$ d\mathcal{G}|_{p}:T_{p}X\rightarrow T_{\mathcal{G}(p)}\mathcal{G}(X). $$ For example, the differential $d\pi|_{y}:T_{y}Y\rightarrow\mathbb{P}^{m}$ of the projection $\pi:Y\rightarrow \mathbb{P}^{m}$ with center at a point $p\notin Y\cup \mathbb{P}^{m}$ is the projection of $T_{y}Y$ to $\mathbb{P}^{m}$ with center at $p$.
Is there any similar result for the morphism $\mathcal{G}$?