Let $k$ be a field, let $n\ge 1$, and let $F\in k[X_0,\dots,X_n]$ be a nonzero [homogeneous polynomial](/page/Homogeneous%20Polynomial) of degree $d\ge 1$. Let
be the projective hypersurface defined by $F$. If $p=[a_0:\cdots:a_n]\in X(k)$ is a smooth point and $a=(a_0,\dots,a_n)\in k^{n+1}\setminus\{0\}$ is the chosen affine representative, then the projective tangent hyperplane to $X$ at $p$ is