Source Latex
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% Fiche format a5 compilée suivant la chaîne:%
% latex file %
% dvips -t a5 -o file.ps file; %
% ps2pdf -sPAPERSIZE=a4 file.ps %
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\documentclass[a5paper,11pt]{book}
\usepackage{amsfonts}\usepackage{amssymb}
\usepackage[french]{babel}
\usepackage{amsmath}
\usepackage[utf8]{inputenc}
\usepackage{graphicx}
\usepackage{epsf}
\usepackage{calc}
\usepackage{array}
\usepackage{multirow}
\usepackage{longtable}
\usepackage{pst-all}
\usepackage{enumerate}
% Raccourcis diverses:
\newcommand{\nwc}{\newcommand}
\nwc{\dsp}{\displaystyle}
\nwc{\ct}{\centerline}
\nwc{\bge}{\begin{equation}}\nwc{\ene}{\end{equation}}
\nwc{\bgar}{\begin{array}}\nwc{\enar}{\end{array}}
\nwc{\bgit}{\begin{itemize}}\nwc{\enit}{\end{itemize}}
\nwc{\bgen}{\begin{enumerate}}\nwc{\enen}{\end{enumerate}}
\nwc{\la}{\left\{}\nwc{\ra}{\right\}}
\nwc{\lp}{\left(}\nwc{\rp}{\right)}
\nwc{\lb}{\left[}\nwc{\rb}{\right]}
\nwc{\bgsk}{\bigskip}
\nwc{\vsp}{\vspace{0.1cm}}
\nwc{\vspd}{\vspace{0.2cm}}
\nwc{\vspt}{\vspace{0.3cm}}
\nwc{\vspq}{\vspace{0.4cm}}
\def\N{{\rm I\kern-.1567em N}}
\def\D{{\rm I\kern-.1567em D}}
\def\No{\N_0}
\def\R{{\rm I\kern-.1567em R}}
\def\C{{\rm C\kern-4.7pt
\vrule height 7.7pt width 0.4pt depth -0.5pt \phantom {.}}}
\def\Q{\mathbb{Q}}
\def\Z{{\sf Z\kern-4.5pt Z}}
\def\epsi{\varepsilon}
\def\vphi{\varphi}
\def\lbd{\lambda}
\def\ga{\gamma}
\def\Cf{\mathcal{C}_f}
\nwc{\tm}{\times}
\nwc{\V}[1]{\overrightarrow{#1}}
\nwc{\zb}{\mbox{$0\hspace{-0.67em}\mid$}}
\nwc{\db}{\mbox{$\hspace{0.1em}|\hspace{-0.67em}\mid$}}
\nwc{\ul}[1]{\underline{#1}}
\nwc{\bgmp}{\begin{minipage}}\nwc{\enmp}{\end{minipage}}
\nwc{\limcdt}[4]{
$\dsp
\lim_{\bgar{ll}\scriptstyle{#1}\vspace{-0.2cm}\\\scriptstyle{#2}\enar}
{#3}={#4}$
}
\headheight=0cm
\topmargin=-1.8cm
\oddsidemargin=-1.7cm
\evensidemargin=-1.7cm
\textwidth=13.2cm
\textheight=18.3cm
% Bandeau en bas de page
\newcommand{\TITLE}{Formules de dérivation}
\author{Y. Morel}
\date{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\pagestyle{empty}
\psset{arrowsize=6pt}
\definecolor{filigray}{gray}{0.82}
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\textcolor{filigray}{\huge\bf xymaths.free.fr}
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\vspace*{-0.9cm}
\ct{\LARGE \bf \TITLE}
\vspd
\noindent
%\bgmp{11.5cm}
{\Large\bf\hspace{-0.3cm}$\bullet$Fonctions de référence}
\newcolumntype{M}[1]{>{\raggedright}m{#1}}
\begin{longtable}{|c|c|c|c|}\hline
&&\multicolumn{2}{c|}{Ensembles de}\\
\raisebox{0.2cm}[.3cm]{Fonction} &
\raisebox{0.2cm}[.3cm]{Dérivée} &
définition&dérivabilité
%\raisebox{0.4cm}[1.2cm]{Ensemble de définition}
%&\raisebox{0.4cm}[1.2cm]{\bgmp{2.25cm}Ensemble de dérivabilité\enmp}
\tabularnewline\hline
\raisebox{0.1cm}[0.6cm]{$k\in\R$ (constante)} &
\raisebox{0.1cm}[0.6cm]{$0$} &
\multirow{3}{*}{$\R$} &
\multirow{3}{*}{$\R$} \\\cline{1-2}
\raisebox{0.1cm}[0.6cm]{$x$} &
\raisebox{0.1cm}[0.6cm]{$1$} & & \\\cline{1-2}
\raisebox{0.1cm}[0.6cm]{$x^n$, $n\in\N$} &
\raisebox{0.1cm}[0.6cm]{$nx^{n-1}$} &&\\\hline
\raisebox{0.2cm}[0.8cm]{$\dfrac{1}{x}$} &
\raisebox{0.2cm}[0.8cm]{$-\dfrac{1}{x^2}$} &
\multirow{2}{*}{$\R^*$} &
\multirow{2}{*}{$\R^*$} \\\cline{1-2}
\raisebox{0.25cm}[0.9cm]{$\dsp\frac{1}{x^n}$,\ $n\in\N$} &
\raisebox{0.25cm}[0.9cm]{$\dsp-n\frac{1}{x^{n+1}}$} & & \\\hline
\raisebox{0.2cm}[0.8cm]{$\sqrt{x}$} &
\raisebox{0.2cm}[0.8cm]{$\dsp\frac{1}{2\sqrt{x}}$} &
\raisebox{0.3cm}[0.8cm]{$\R_+=[0;+\infty[$} &
\raisebox{0.3cm}[0.8cm]{$\R_+^*=]0;+\infty[$}\\\hline
\raisebox{0.1cm}[0.6cm]{$\sin(x)$} &
\raisebox{0.1cm}[0.6cm]{$\cos(x)$} &
\multirow{3}{*}{$\R$} &
\multirow{3}{*}{$\R$} \\\cline{1-2}
\raisebox{0.1cm}[0.6cm]{$\cos(x)$} &
\raisebox{0.1cm}[0.6cm]{$-\sin(x)$}
&&\\\cline{1-2}
%\raisebox{0.3cm}[1cm]{$\dsp\tan(x)=\frac{\sin(x)}{\cos(x)}$}&
%\raisebox{0.3cm}[1cm]{$\dsp 1+\tan^2(x)=\frac{1}{\cos^2(x)}$}&
%\raisebox{0.3cm}[1cm]{$\dsp\R\setminus\la \frac{\pi}{2}+k\pi;\,k\in\Z\ra$}&
%\raisebox{0.3cm}[1cm]{$\dsp\R\setminus\la \frac{\pi}{2}+k\pi;\,k\in\Z\ra$}
%\\\hline
\raisebox{0.1cm}[0.6cm]{$e^x$} &
\raisebox{0.1cm}[0.6cm]{$e^x$} & & \\\hline
\raisebox{0.3cm}[1cm]{$\ln(x)$} &
\raisebox{0.3cm}[1cm]{$\dsp\frac{1}{x}$}&
\raisebox{0.3cm}[1cm]{$\R_+^*=]0;+\infty[$} &
\raisebox{0.3cm}[1cm]{$\R_+^*=]0;+\infty[$}
\\\hline
\end{longtable}
%\vspace{-1.cm}
\noindent
%\fbox{
\bgmp[t]{7.5cm}
{\Large\bf\hspace{-0.3cm}$\bullet$Opérations sur les dérivées}
%\noindent
%$u$ et $v$ désignent deux fonctions quelconques, définies
%respectivement sur $D_u$ et $D_v$, dérivables sur
%$D'_u$ et $D'_v$.
%
%On note de plus $D_v^*=\la x\in D_v, \mbox{ tel que, } v(x)\not=0\ra$.
%\vspd
%\begin{longtable}{|c|c|c|c|}\hline
\begin{longtable}{|c|c|}\hline
\raisebox{0.1cm}[0.5cm]{Fonction} &
\raisebox{0.1cm}[0.5cm]{Dérivée}
%&\raisebox{0.4cm}[1.2cm]{\bgmp{2.25cm}Ensemble de définition\enmp}
%&\raisebox{0.4cm}[1.2cm]{\bgmp{2.25cm}Ensemble de dérivabilité\enmp}
\tabularnewline\hline
\raisebox{0.1cm}[0.7cm]{$ku$, $k\in\R$} &
\raisebox{0.1cm}[0.7cm]{$ku'$}
%& \raisebox{0.1cm}[0.7cm]{$D_u\cap D_v$} &
%\raisebox{0.1cm}[0.7cm]{$D'_u\cap D'_v$}
\\\hline
\raisebox{0.1cm}[0.7cm]{$u+v$} &
\raisebox{0.1cm}[0.7cm]{$u'+v'$}
%& \raisebox{0.1cm}[0.7cm]{$D_u\cap D_v$} &
%\raisebox{0.1cm}[0.7cm]{$D'_u\cap D'_v$}
\\\hline
\raisebox{0.1cm}[0.7cm]{$uv$} &
\raisebox{0.1cm}[0.7cm]{$u'v+uv'$}
%& \raisebox{0.1cm}[0.7cm]{$D_u\cap D_v$} &
%\raisebox{0.1cm}[0.7cm]{$D'_u\cap D'_v$}
\\\hline
\raisebox{0.3cm}[1.1cm]{$\dsp\frac{u}{v}$} &
\raisebox{0.3cm}[1.1cm]{$\dsp\frac{u'v-uv'}{v^2}$}
%& \raisebox{0.3cm}[1.1cm]{$D_u\cap D_v^*$}&
%\raisebox{0.3cm}[1.1cm]{$D'_u\cap D_v^{'*}$}
\\\hline
\raisebox{0.1cm}[0.7cm]{$u\circ v$} &
\raisebox{0.1cm}[0.7cm]{$v'\tm u'\circ v$}
%&&
\\
\raisebox{0.1cm}[0.7cm]{$u(v(x))$} &
\raisebox{0.1cm}[0.7cm]{$v'(x)\tm u'(v(x))$}
\\\hline
\end{longtable}
\enmp%}\fbox{
\bgmp[t]{5.5cm}
{\Large\bf\hspace{-0.3cm}$\bullet$Compositions usuelles}
%\vspq
%$u$ est une fonction quelconque définie et dérivable sur un intervalle
%$I$
%(et ne s'annulant pas sur $I$ pour les quotients, racines carrées et
%logarithmes).
\begin{longtable}{|c|c|}\hline
\raisebox{0.1cm}[0.5cm]{Fonction} &
\raisebox{0.1cm}[0.5cm]{Dérivée}
\tabularnewline\hline
\raisebox{0.05cm}[0.5cm]{$u^n$, $n\in\Z$, $n\not=0$} &
\raisebox{0.05cm}[0.5cm]{$nu'u^{n-1}$} \\\hline
\raisebox{0.15cm}[0.8cm]{$\dsp\frac{1}{u^n}$, $n\in\Z$, $n\not=0$} &
\raisebox{0.15cm}[0.8cm]{$\dsp -\frac{nu'}{u^{n+1}}$} \\\hline
$\sqrt{u}$ & $\dsp\frac{u'}{2\sqrt{u}}$ \\\hline
\raisebox{0.1cm}[0.6cm]{$\sin(u)$} &
\raisebox{0.1cm}[0.6cm]{$u'\cos(u)$} \\\hline
\raisebox{0.1cm}[0.6cm]{$\cos(u)$} &
\raisebox{0.1cm}[0.6cm]{$-u'\sin(u)$} \\\hline
%\raisebox{0.3cm}[1cm]{$\dsp\tan(u)=\frac{\sin(u)}{\cos(u)}$}&
%\raisebox{0.3cm}[1cm]{$\dsp u'\lb 1+\tan^2(u)\rb=\frac{u'}{\cos^2(u)}$}
%\\\hline
\raisebox{0.cm}[0.4cm]{$e^u$} & $u'e^u$\\\hline
\raisebox{0.15cm}[0.8cm]{$\ln(u)$} &
\raisebox{0.15cm}[0.8cm]{$\dsp\frac{u'}{u}$}
\\\hline
\end{longtable}
\enmp
%}
\end{document}
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