Tangent Spaces and Morphisms of Affine Varieties In page 205 of "Algebraic Curves, Algebraic Manifolds and Schemes" by Shokurov and Danilov, the tangent space $T_x X$ of an affine variety $X$ at a point $x \in X$ is defined as the subspace of $K^n$, where $K$ is the underlying field, such that $\xi \in T_x X$ if $(d_x g)(\xi)=0$ for any $g \in I$, where $I$ is the ideal of $K[T_1,\cdots,T_n]$ that defines $X$ and by definition $(d_x g)(\xi)=\sum_{i=1}^n \frac{\partial g}{\partial T_i}(x) \xi_i$, where partial derivatives are formal. So far so good. Next, it is mentioned, that if $f:X \rightarrow Y$ is a morphism of affine varieties, then we obtain a well-defined map $d_x f : T_x X \rightarrow T_{f(x)} Y$. How is this mapped defined and why is it well-defined?

This is explained in Shafarevich, p.88. Let $f(x)=y.$ The intrinsic and natural way to define this map is to use the isomorphisms $T_xX\cong (\mathfrak m_x/\mathfrak m_x^2)^*$ and $T_yY\cong(\mathfrak m_y/\mathfrak m_y^2)^*$ and the fact that $f$ defines a pullback on regular functions $f^*:k[Y]\to k[X].$ The map $f^*$ induces a map $\mathfrak m_y\to\mathfrak m_x$ which factors to give $d_x^*:\mathfrak m_y/\mathfrak m_y^2\to\mathfrak m_x/\mathfrak m_x^2.$
