Let $X$ be a scheme. For any point $x \in X$, we define the Zariski tangent space $T_x$ to $X$ at $x$ to be the dual of the $k(x)$-vector space $\mathfrak m_x/\mathfrak m_x^2$. Now assume that $X$ is a scheme over a field $k$, and let $k[\varepsilon]/\varepsilon^2$ be the ring of dual numbers over $k$. Show that to give a $k$-morphism of $\text{Spec} (k[\varepsilon]/\varepsilon^2)$ to $X$ is equivalent to giving a point $x \in X$, rational over $k$ (i.e., such that $k(x) =k$), and an element of $T_x$.
$\text{Spec} (k[\varepsilon]/\varepsilon^2)$ is a $k$-scheme in the above Problem 2.8 Hartshorne. In order to solve the problem, do we have to assume that the $k$-scheme structure on it is induced by natural ring homomorphism from $k$ to $k[\varepsilon]/\varepsilon^2$ sending an element of the field $k$ to it's equivalence class modulo $\varepsilon^2$?