Your question is closely related to this one, and I slightly modified the answer there to get
\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{pgfplots}
\usepgfplotslibrary{fillbetween}
\usetikzlibrary{calc,decorations.markings}
\begin{document}
\begin{tikzpicture}[scale=1.4,tangent at/.style={% cf. https://tex.stackexchange.com/questions/25928/how-to-draw-tangent-line-of-an-arbitrary-point-on-a-path-in-tikz/25940#25940
decoration={ markings,
mark =at position #1 with {\draw[purple,-latex](-1,0) -- (1,0);},
}, decorate
}]
\clip (-3,-2) rectangle (3,2);
\coordinate (A) at (130:2 and 1);
\draw[black,thick,name path=elli] (0,0) ellipse (2 and 1);
\node[label=above:$A$,circle,fill,inner sep=0.05cm] at (A){};
\path[name path=clip] (130:1 and 0.5) -- (130:2.2 and 1.1);
\path [%draw,blue,
name path=middle arc,
intersection segments={
of=elli and clip,
sequence={A1}
},
postaction={tangent at/.list={0}}];
\end{tikzpicture}
\end{document}

This is a general way that works for curves whose parametrization you do not know.
Of course, for an ellipse this is an overkill. You can just compute the tangent analytically.
\documentclass[tikz,border=3.14mm]{standalone}
\begin{document}
\begin{tikzpicture}[scale=1.4]
\clip (-3,-2) rectangle (3,2);
\coordinate (A) at (130:2 and 1);
\draw[black,thick] (0,0) ellipse (2 and 1);
\node[label=above:$A$,circle,fill,inner sep=0.05cm] at (A){};
\draw[blue] (A) -- ++ ({-2*sin(130)},{cos(130)})
(A) -- ++ ({2*sin(130)},{-cos(130)});
\node[above]{$\gamma(t)=\bigl(2\,\cos(t),\sin(t)\bigr)$};
\node[below]{$\dot\gamma(t)=\bigl(-2\,\sin(t),\cos(t)\bigr)$};
\end{tikzpicture}
\end{document}
