\draw slope triangle with pgfplots using relative coordinates

pgfplotstikz-pgf

I want to use the \draw command in conjunction with relative coordinates in order to plot a slope triangle underneath a tangent to a graph, but the following syntax does not yield the desired result:

\draw [color=red] (-1,4) -- ++(0,-1) -- ++(0.5,0);

Instead, the result I obtain is the following:

enter image description here

Here is the complete code:

\documentclass{article}
\usepackage[utf8]{inputenc}

\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[ngerman]{babel}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{positioning,intersections,shapes.geometric,automata,math}

\usepackage{pgfplots}
\pgfplotsset{width=10cm,compat=newest}
\usepgfplotslibrary{external}
\tikzexternalize

\begin{document}

\begin{tikzpicture}
[declare function={
f(\x) = \x^3 - 5*\x;
derf(\x) = 3*\x^2 - 5;
g(\x,\y) = derf(\y)*(\x-\y) + f(\y);
}]
\begin{axis}[
    axis x line=center,
    axis y line=center,
    xtick=\empty,
    ytick=\empty,
    xlabel={$x$},
    ylabel={$y$},
    xlabel style={below right},
    ylabel style={above left},
    xmin=-3.5,
    xmax=3.5,
    ymin=-15,
    ymax=15
    ]
    
    \addplot [domain=-3:3,smooth,thick, color=black, name path=p1] {f(x)} node[above right, xshift=-9mm] {$y=f(x)$};
    
    \tikzmath{\z1=-2.6;}
    \addplot [only marks,mark=*] coordinates { (\z1,{f(\z1)}) };
    \addplot [domain=-3.1:-2.1, color=black] { g(x,\z1) };
    \draw [dashed] (\z1,0) -- (\z1,{f(\z1)});
    \draw (\z1,2pt) -- (\z1,-2pt) node[above] {$x_1$};
   
    \tikzmath{\z2=-1;}
    \addplot [only marks,mark=*] coordinates { (\z2,{f(\z2)}) };
    \addplot [domain=-2.3:0.3, color=black] { g(x,\z2) };
    \draw [dashed] (\z2,0) -- (\z2,{f(\z2)});
    \draw (\z2,2pt) -- (\z2,-2pt) node[below] {$x_2$};
    \draw [color=red] (-1,4) -- ++(0,-1) -- ++(0.5,0);
    
    \tikzmath{\z3=1.8;}
    \addplot [only marks,mark=*] coordinates { (\z3,{f(\z3)}) };
    \addplot [domain=0.8:2.8, color=black] { g(x,\z3) };
    \draw [dashed] (\z3,0) -- (\z3,{f(\z3)});
    \draw (\z3,2pt) -- (\z3,-2pt) node[above] {$x_3$};
\end{axis}
\end{tikzpicture}

\end{document}

I am quite new to Tikz and Pgfplots and perhaps I just made a silly mistake. In any case, I would be really grateful if someone would help me fix this.

Apart from this I would also really appreciate any suggestions or advice to improve the code, simplify it and clean it up where possible. For example, I was wondering if the three blocks of code to plot the tangent lines could be replaced by just one block, using a common variable taking three different values instead of defining the three variables \z1, \z2, \z3separately.

Many thanks in advance!

Best Answer

Your code does not compile, so I offer only a one line solution instead of compilable code:

\draw [red] (-1,4) -- ++(axis direction cs:0,-1) -- ++(axis direction cs:0.5,0);  %ok now

https://tex.stackexchange.com/a/332603/8650

Edit:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{arrows.meta}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\begin{document}

\begin{tikzpicture}[
    declare function={
    f(\x) = \x^3 - 5*\x;
    derf(\x) = 3*\x^2 - 5;
    g(\x,\xzero) = derf(\xzero)*(\x-\xzero) + f(\xzero);
    x1=-2.6;
    x2=-1;
    x3=1.8;
    }]
    
\begin{axis}[
    axis x line=center, axis y line=center,
    xtick=\empty, ytick=\empty,
    xlabel={$x$}, ylabel={$y$},
    xlabel style={below right}, ylabel style={above left},
    xmin=-3.5, xmax=3.5,
    ymin=-15, ymax=15,
    clip=false,
    xtick={x1, x2, x3}, xticklabel=\empty
    ]
        
    \draw[dashed] (x1,0) node[above] {\footnotesize $x_1$} -- (x1,{f(x1)});
    \addplot[domain=-3.1:-2.1] { g(x,x1) };
    \draw plot[mark=*, mark size=1.5pt] coordinates { (x1,{f(x1)}) };

    \draw[dashed] (x2,0) node[below] {\footnotesize $x_2$} -- (x2,{f(x2)});
    \draw [red] (x2,{f(x2)}) -- ++(axis direction cs:0,-1) -- ++(axis direction cs:{-1/derf(x2)},0);
    \addplot[domain=-1.8:-0.2] { g(x,x2) };
    \draw plot[mark=*, mark size=1.5pt] coordinates { (x2,{f(x2)}) };
    
    \draw[dashed] (x3,0) node[above] {\footnotesize $x_3$} -- (x3,{f(x3)});
    \addplot[domain=1.2:2.3] { g(x,x3) };
    \draw plot[mark=*, mark size=1.5pt] coordinates { (x3,{f(x3)}) };
   
    \addplot [domain=-3:3,smooth,thick] {f(x)} node[above] {\footnotesize $y=f(x)$};

\end{axis}
\end{tikzpicture}
\end{document}

Graph

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