Derivative Vector Function


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% File name: derivative-vector-function.tex
% Description: 
% A geometric representation of the steps one and two used 
% (from the four step rule)
% to compute the derivative of a vector function is shown.
% Date of creation: October, 10th, 2021.
% Date of last modification: October, 9th, 2022.
% Author: Efraín Soto Apolinar.
% Source: page 211 of the 
% Glosario Ilustrado de Matem\'aticas Escolares.
% Terms of use:
% According to
% Your commitment to the terms of use is greatly appreciated.
	% Component functions of the vector function
	\tikzmath{function equis(\t) {return cos((\t) r);};}
	\tikzmath{function ye(\t) {return sin((\t) r);};}
	\tikzmath{function zeta(\t) {return 0.25+sqrt(\t);};}
	% Evaluated at $t = \ti$
	% Coordinate axis
	\draw[thick,->] (-1.25,0,0) -- (1.5,0,0) node[below left] {$x$}; % Eje x
	\foreach \x in {-1,1}
		\draw[thick] (\x,0,0.05) -- (\x,0,-0.05) node [below] {$\x$};
	\draw[thick,->] (0,-1.25,0) -- (0,1.5,0) node[right] {$y$}; % Eje y
	\foreach \y in {-1,1}
		\draw[thick] (0,\y,0.05) -- (0,\y,-0.05) node [below] {$\y$};
	\draw[thick] (0,0,-0.25) -- (0,0,0.75); % Eje z (Primera parte)
	% Graph of the function $\vec{r}(t)$
	\draw[red,thick,->] plot[domain=\ti:\tf,smooth,variable=\t] ({\r*equis(\t)},{\r*ye(\t)},{\r*zeta(\t)});
	\node[red,above] at (\xf,\yf,\zf+0.25) {$\vec{r(t)}$};
	% $z$ axis
	\draw[thick,->] (0,0,0.75) -- (0,0,\zf+0.5) node[above] {$z$}; % Eje z
	\foreach \z/\posicion in {1/left}
		\draw[thick] (0,0.05,\z) -- (0,-0.05,\z) node [\posicion] {$\z$};
	% Node $\vec{r}(t)$
	\draw[blue,thick,->] (0,0,0) -- (\xtcero,\ytcero,\ztcero) node[sloped,below,near end] {\footnotesize$\vec{r}(t)$};	
	% Node $\vec{r}(t + \Delta t)$
	\draw[blue,thick,->] (0,0,0) -- (\xtuno,\ytuno,\ztuno) node[sloped,above,near end] {\footnotesize$\vec{r}(t + \Delta t)$};
	% Node $\Delta r$
	\draw[blue,thick,->] (\xtcero,\ytcero,\ztcero) -- (\xtuno,\ytuno,\ztuno) node[midway,sloped,above] {\footnotesize$\Delta\vec{r}$};	

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