Let $\Bbbk$ be a field (I am interested in $\Bbbk=\mathbb C$). Let $X$ be a $\Bbbk$-scheme of finite type and let $D\hookrightarrow X$ be an effective Cartier divisor. I want to justify or disporve the following intuition \begin{equation} T_xD\subsetneq T_xX,\quad \textrm{for all }x:\mathrm{Spec}(\Bbbk)\to D \end{equation} where $T_xD$ is the set of morphisms $\mathrm{Spec}(\Bbbk[t]/t^2)\to D$ lifting $x:\mathrm{Spec}(\Bbbk)\to D$.
If the above statement is true, then how about the following
For any $\Bbbk$-point of $D$, there exsits a curve $C\hookrightarrow X$ such that: (1) $C$ is not in $D$, and (2) $x\in C$ is a Cartier divisor.
The first statement about tangent vectors is disproved by David. Singular curves in $\mathbb A^2$ and their singular points are counterexamples.