The Jacobian criterion does not work on non-closed points of a $\mathbb{K}$-scheme locally of finite type; or it does not work as we know it!
Let $X$ be a scheme over a field $\mathbb{K}$, locally of finite type; that is:
\begin{gather}
\forall P\in X,\,\exists U\subseteq X\,\text{open,}\,t_1,\dots,t_n\,\text{indeterminates,}\,f_1,\dots,f_r\in\mathbb{K}[t_1,\dots,t_n]=R,\\(f_1,\dots,f_r)=I:
P\in U\simeq\operatorname{Spec}R_{\displaystyle/I}\cong V(I)\subseteq\mathbb{A}^n_{\mathbb{K}}\,\text{affine closed subscheme}.
\end{gather}
By hypothesis
\begin{equation}
\forall P\in U\subseteq X,\,T_PX\cong T_PU\cong T_PV(I)\leq T_P\mathbb{A}^n_{\mathbb{K}}=T_{\mathfrak{m}_P}\mathcal{O}_{\mathbb{A}^n_{\mathbb{K}},P}=\left(\mathfrak{m}_{P\displaystyle/\mathfrak{m}_P^2}\right)^{\vee}\cong\kappa(P)^n
\end{equation}
where $\mathfrak{m}_P$ is the maximal ideal of the local ring $\mathcal{O}_{\mathbb{A}^n_{\mathbb{K}},P}$ and $\kappa(P)$ is the relavant residue field.
If $P$ is a closed point of $X$, then $P$ is a closed point in $U$; let $\mathfrak{p}$ be the maximal ideal of $R$ corresponding to $P$, and let $\varphi:\mathbb{K}[s_1,\dots,s_r]\to\mathbb{K}[t_1,\dots,t_n]$ the morphism of $\mathbb{K}$-algebras such that
\begin{equation}
\forall i\in\{1,\dots,r\},\,\varphi(s_i)=f_i;
\end{equation}
then:
\begin{gather*}
\operatorname{coker}\varphi=R_{\displaystyle/I},\\
\varphi^{*}:\mathbb{A}^n_{\mathbb{K}}\to\mathbb{A}^r_{\mathbb{K}},\\
\ker\varphi^{*}=V(I);
\end{gather*}
let $\varphi_0=\varphi_{\varphi^{*}(P)}:\mathbb{K}[s_1,\dots,s_r]_{(s_1,\dots,s_r)}\to\mathbb{K}[t_1,\dots,t_n]_{\mathfrak{p}}$, by definition:
\begin{equation}
(d_P\varphi_0)^{\vee}:\left(T_O\mathbb{A}^r_{\mathbb{K}}\right)^{\vee}\to \left(T_P\mathbb{A}^n_{\mathbb{K}}\right)^{\vee}
\end{equation}
and in particular:
\begin{equation}
\left(T_PV(I)\right)^{\vee}=\operatorname{coker}(d_P\varphi_0)^{\vee}\Rightarrow T_PV(I)=\ker d_P\varphi_0;
\end{equation}
where $(d_P\varphi_0)^{\vee}$ is a $\mathbb{K}$-linear map and $T_P\mathbb{A}^n_{\mathbb{K}}\cong\kappa(P)^n\cong\mathbb{K}^m$ by Hilbert's (Strong) Nullstellensatz.
REMARK1: If $P$ is not a closed point of $X$, that is a closed point of $\mathbb{A}^n_{\mathbb{K}}$, we can't apply the Hilbert (Strong) Nullstellensatz.
By Hilbert's Basis Theorem, $\mathfrak{p}$ is a finite generated $\mathbb{K}$-vector space, therefore:
\begin{gather}
\mathfrak{p}=(e_1,\dots,e_m)\\
\forall i\in\{1,\dots,r\},\,(d_P\varphi_0)^{\vee}(\overline{s_i})=\overline{f_i}=\sum_{j=1}^ma_i^j\overline{e_j},\,\text{where:}\,a_i^j\in\mathbb{K};
\end{gather}
but every $a_i^j$ no makes sense as the formal derivation of $f_i$ with respect to the element $e_j$ computed at $P$, unless $P$ is a closed $\mathbb{K}$-point of $X$! By a base change, we can define $\overline{\varphi}:\mathbb{F}[s_1,\dots,s_r]\to\mathbb{F}[t_1,\dots,t_n]$ where $\mathbb{F}=\kappa(P)$, we can repeat the same reasoning described for $\varphi$ and we can prove that:
\begin{equation}
T_P(V(I)/\operatorname{Spec}\mathbb{F})=\ker d_P\overline{\varphi}_0
\end{equation}
where the notation is clear.
Because $P$ is a closed $\mathbb{F}$-point of $\mathbb{A}^n_{\mathbb{F}}$, then:
\begin{equation}
\exists\alpha_1,\dots,\alpha_n\in\mathbb{F}\mid\mathfrak{p}=(t_1-\alpha_1,\dots,t_n-\alpha_n)
\end{equation}
and therefore, via a formal Taylor series of the $f_i$'s
\begin{gather}
\forall i\in\{1,\dots,r\},\,(d_P\varphi_0)^{\vee}(\overline{s_i})=\overline{f_i}=\dots=\sum_{j=1}^n\frac{\partial f_i}{\partial t_j}\bigg|_{(t_1-\alpha_1,\dots,t_n-\alpha_n)}\left(\overline{t_j-\alpha_j}\right)
\end{gather}
that is $T_P(V(I)/\operatorname{Spec}\mathbb{F})$ is the kernel of the linear map from $\mathbb{F}^r$ to $\mathbb{F}^n$ described from the Jacobian matrix of the $f_i$'s with respect to $t_j$'s, with entries in $\mathbb{F}$ and valued in $P$.
I repeat, the definition 12.2.6 and it is completed by exercise 12.2.H, it is true that this definition is intricated and in apparence is unsatisfactory; but this definition is completely right and it works on $\mathbb{K}$-schemes locally of finite type.
REMARK2: In general $T_PV(I)$ and $T_P(V(I)/\operatorname{Spec}\mathbb{F})$ are not isomorphic as $\mathbb{F}$-vector spaces; see exercise 6.3 from Görtz and Wedhorn - Algebraic Geometry I, Schemes With Examples and Exercises.
EDIT: The enumeration is refered to December 29 2015 FOAG version.
