f(x) = frac{x^4}{12}-frac{2x^3}{3}+frac{3 x^2}{2}

documentclass[tikz,border=3.14mm]{standalone}
usepackage{pgfplots}
pgfplotsset{compat=1.16}
usepgfplotslibrary{fillbetween}
makeatletter
defparsenode[#1]#2pgf@nil{%
    tikzset{label node/.style={#1}}
    defnodetext{#2}
}

tikzset{
    add node at x/.style 2 args={
        name path global=plot line,
        /pgfplots/execute at end plot visualization/.append={
            begingroup
            @ifnextchar[{parsenode}{parsenode[]}#2pgf@nil
            path [xshift=-0.1pt,name path global = position line #1-1]
            ({axis cs:#1,0}|-{rel axis cs:0,0}) --
            ({axis cs:#1,0}|-{rel axis cs:0,1});
            path [xshift=0.1pt, name path global = position line #1-2]
            ({axis cs:#1,0}|-{rel axis cs:0,0}) --
            ({axis cs:#1,0}|-{rel axis cs:0,1});
            path [
            name intersections={
                of={plot line and position line #1-1},
                name=left intersection
            },
            name intersections={
                of={plot line and position line #1-2},
                name=right intersection
            },
            label node/.append style={pos=1}
            ] (left intersection-1) -- (right intersection-1)
            node [label node]{nodetext};
            endgroup
        }
    }
}
makeatother

begin{document}
    begin{tikzpicture}[>=latex]
    begin{axis}[
    xtick={-2,-1,...,5},
    xticklabel=empty,
    ytick={-2,-1,...,5},
    yticklabel=empty, grid=major,
    axis x line=center,
    axis y line=center,
    xlabel={$x$},
    ylabel={$y$},
    xlabel style={below},
    ylabel style={left},
    xmin=-2,
    xmax=5,
    ymin=-2,
    ymax=5,
    tangent/.style={
        add node at x={#1}{
            [
            sloped,
            append after command={(tikzlastnode.west) edge [thick,black] (tikzlastnode.east)},
            minimum width=0.2textwidth
            ]
        }
    }]
    addplot[color=blue,smooth,samples=501, thick,tangent/.list={1,3},domain=-5:5] {(3* x^2)/2 - (2 *x^3)/3 + x^4/12};
    end{axis}
    end{tikzpicture}
end{document}