I am reading Shaferevich's Basic Algebraic Geometry. In chapter 2, Section 1, he defined tangent space at a point of a quasiprojective variety in the following way:
A quasiprojective variety is open subset of a closed set in some projective space. A quasiprojective variety is called an affine variety if it is isomorphic to some affine closed set. Let $X$ be a quasiprojective variety and $x\in X$. We define the tangent space to $X$ at $x$, $\Theta_{X,x}:=(\mathcal m_x/\mathcal m_x^2)^*$, the dual vector space, where $\mathcal m_x$ is the maximal ideal of the local ring $\mathcal O_{X,x}$. Let $Y\subseteq X$ is a subvariety i.e., $Y$ is also a quasiprojective variety and $x\in Y$ then my question is:
$\textbf{ Why $\Theta_{Y,x}\subseteq \Theta_{X,x}$}$ ?
I have proved this for $X$ and $Y$ both being affine varieties. I also know that for any quasiprojective variety $X$, $\Theta_{X,x}=\Theta_{U,x}$ where $U$ is an affine neighbourhood of $x$ in $X$. Can this two results be used to prove the above claim? Please help me. Thank you.