Questions about tangent spaces of $A^n$.

Questions about tangent spaces of $A^n$.

Let $A^n=k\times \cdots \times k$ be the affine $n$-space, $k$ is a field.
Define $d_xf = \sum_{i=1}^n \frac{\partial f}{\partial T_i}(x)(T_i-x_i)$,
where $x=(x_1, \ldots, x_n) \in A^n$.
Suppose that $X \subset A^n$ is defined by polynomials $f(T)=f(T_1,
\ldots, T_n)$. Define $$ \operatorname{Tan}(X)_x = \{x \in A^n \mid
d_xf(x) = 0, \forall f(T) \in I(X) \}. $$
Suppose that $I(X)$ is generated by $f_1(T), \ldots, f_m(T)$. Then each
element in $I(X)$ is of the form $a_1f_1+\cdots +a_nf_n$ for some $a_i \in
I(X)$. How to show that $d_xf_1, \ldots, d_xf_m$ generated by ideal of
$\operatorname{Tan}(X)_x$? Thank you very much.
I think that the ideal of $\operatorname{Tan}(X)_x$ is $$ \{f \mid f(x) =
0, \forall x \in \operatorname{Tan}(X)_x \}. $$ If $x \in
\operatorname{Tan}(X)_x$, then $d_xf(x)=0$ for all $f\in I(X)$. That is,
$d_xf(x)=0$ for all $f$ which are of the form $a_1f_1+\cdots a_mf_m$.
Therefore the ideal of $\operatorname{Tan}(X)_x$ is contained in the ideal
generated by $d_xf_1, \ldots, d_xf_m$. If $f$ is generated by $d_xf_1,
\ldots, d_xf_m$, we also have $f(x)=0$ for all $x \in
\operatorname{Tan}(X)_x$. Hence $x$ is in the ideal of
$\operatorname{Tan}(X)_x$. Is this correct? Thank you very much.