Let $X$ be the union of the $3$ coordinate axes in $\mathbb{A}^{3}$. I want to compute the tangent space of $X$ at every $p \in \mathbb{A}^{3}$.
One can check that the only singular point of $X$ is the origin so if $p$ is the origin then the tangent space of $X$ at the origin is equal to $\mathbb{A}^{3}$.
Question: how do we express the answer if $p \in \mathbb{A}^{3} \setminus \{(0,0,0)\}$? Would it be correct to say:
Note that $I(X)=(xy,xz,yz)$ so if $p=(a,b,c) \in \mathbb{A}^{3}$ then:
$T_{p}(X):=\{(x_{1},x_{2},x_{3}) \in \mathbb{A}^{3}: \sum_{i=1}^{3} \frac{\partial f}{\partial x_{i}}(a,b,c)(x_{i})=0\ \text{for every f in I(X)}\}$
In case $p$ is not the origin, is there a "nice" way to express $T_{p}(X)$ other than the above?