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\fancyhead[L]{\textbf{Prof Rachid }}
\fancyhead[C]{\textbf{\Large Série : L'Exponentielle (Calcul Intensif)}}
\fancyhead[R]{\textbf{ 2 Bac SVT+PC }}
\fancyfoot[L]{\textbf{Année 2025-2026}}
\fancyfoot[C]{\thepage}
\fancyfoot[R]{\textbf{ Lycée }}

\begin{document}

\begin{tcolorbox}[colback=white, colframe=exgreen, title=OBJECTIF : AUTOMATISMES DÉRIVATION ET LIMITES, fonttitle=\bfseries]
Pour chacune des fonctions suivantes, calculer les limites aux bornes de $D_f$ et déterminer la fonction dérivée $f'$.
\end{tcolorbox}

\begin{multicols}{4}
\begin{enumerate}
    \item $e^x - x$
    \item $xe^x$
    \item $(x^2-1)e^x$
    \item $\frac{e^x}{x}$
    \item $e^{2x} - 3e^x + 2$
    \item $\frac{1}{e^x + 1}$
    \item $e^{-x^2}$
    \item $(x+1)e^{-x}$
    \item $x - e^x$
    \item $\frac{e^x-1}{e^x+1}$

    \item $e^{\frac{1}{x}}$
    \item $x^2 e^x$
    \item $\frac{x}{e^x}$
    \item $e^{3x-2}$
    \item $\sqrt{e^x}$
    \item $(e^x - 2)^2$
    \item $x e^{-x^2}$
    \item $\frac{e^x}{x^2+1}$
    \item $\ln(e^x + 1)$
    \item $e^x + e^{-x}$

    \item $x^2 - e^x$
    \item $e^{2x} - e^x$
    \item $\frac{e^x}{x-1}$
    \item $(1-x)e^x$
    \item $e^{\sqrt{x}}$
    \item $x^3 e^{-x}$
    \item $\frac{1-e^x}{x}$
    \item $e^{x^2+x+1}$
    \item $(x^2+x)e^{-x}$
    \item $\frac{e^{2x}}{x}$

    \item $e^{x} \ln(x)$
    \item $2e^x - x^2$
    \item $\frac{e^x + e^{-x}}{2}$
    \item $(x-3)^2 e^x$
    \item $e^{-x} \sin(x)$
    \item $\frac{x^2-1}{e^x}$
    \item $e^{1-x}$
    \item $\ln(1+e^{-x})$
    \item $e^{x} + \frac{1}{e^x}$
    \item $x e^{1/x}$

    \item $e^{\cos x}$
    \item $(e^x+1)^3$
    \item $\frac{1}{e^x-1}$
    \item $x^n e^x$
    \item $e^{x} - \sqrt{x}$
    \item $\frac{x+e^x}{x-e^x}$
    \item $e^{x^2-4}$
    \item $(2x+1)e^{2x}$
    \item $\frac{e^x}{\sqrt{x}}$
    \item $e^{x} + x^3 - 1$

    \item $e^{e^x}$
    \item $x - 1 + e^{-x}$
    \item $\frac{e^{2x}-1}{e^x}$
    \item $(x^2-x+1)e^x$
    \item $\sqrt{1+e^x}$
    \item $\frac{x}{1+e^{2x}}$
    \item $e^{-1/x}$
    \item $x \ln(e^x + 1)$
    \item $e^{2x} - 4e^x$
    \item $\frac{e^x}{x+1}$

    \item $e^{x+ \ln x}$
    \item $(x-1)e^{-x} + 2$
    \item $\frac{e^x-x}{e^x}$
    \item $e^{x^3}$
    \item $x^2 e^{-x} + x$
    \item $\frac{e^x}{\ln x}$
    \item $e^{1/x^2}$
    \item $(x+1)^2 e^{-x}$
    \item $\ln(e^x - 1)$
    \item $\frac{e^x+x}{x^2}$

    \item $e^{\tan x}$
    \item $e^{2x} \sqrt{x}$
    \item $\frac{1-x}{e^x}$
    \item $x^x = e^{x \ln x}$
    \item $e^x(x-1)^2$
    \item $\frac{e^x+2}{e^{2x}}$
    \item $e^{-x} \ln(x)$
    \item $\sqrt{e^{2x}+1}$
    \item $x^2 + e^{-x}$
    \item $\frac{\ln(e^x)}{x}$

    \item $e^{\sin^2 x}$
    \item $(x^3-1)e^x$
    \item $\frac{e^x}{x^n}$
    \item $e^x + \cos x$
    \item $x(e^x-1)$
    \item $\frac{x+1}{e^x-1}$
    \item $e^{-x} + \sqrt{x}$
    \item $x^2 e^{1/x}$
    \item $\frac{e^x}{1-e^x}$
    \item $e^{2x-x^2}$

    \item $\ln|e^x - 2|$
    \item $x^2 e^{-x^2}$
    \item $\frac{e^x - e^{-x}}{e^x + e^{-x}}$
    \item $(2-x)e^x + x$
    \item $e^{\sqrt{x^2+1}}$
    \item $\frac{x}{e^x+x}$
    \item $e^x (1 - e^x)$
    \item $\frac{1}{x e^x}$
    \item $\ln(e^{2x} + e^x + 1)$
    \item $x e^x - e^x + 1$

        \item $f_{141}(x) = \frac{e^x - e^{-x}}{2}$
        \item $f_{142}(x) = \ln(x^2 e^x + 1)$
        \item $f_{143}(x) = \frac{x^2 + 1}{e^x}$
        \item $f_{144}(x) = e^{x} \sqrt{\ln(x)}$
        \item $f_{145}(x) = \ln(1 - e^{-x})$
        \item $f_{146}(x) = \frac{e^{2x} + 3}{e^x - 1}$
        \item $f_{147}(x) = (x^2 - 2x + 2)e^x$
        \item $f_{148}(x) = e^{x} \cos(e^x)$
        \item $f_{149}(x) = \ln(e^x + \sqrt{e^{2x}-1})$
        \item $f_{150}(x) = \frac{x}{e^x - x}$
        \item $f_{151}(x) = (e^x - x) \ln(e^x - x)$
        \item $f_{152}(x) = \sqrt{e^{2x} + e^x + 1}$
        \item $f_{153}(x) = \frac{1 + \ln(x)}{e^x}$
        \item $f_{154}(x) = e^{-x} \ln(1+e^x)$
        \item $f_{155}(x) = \frac{e^x}{x^2 \ln(x)}$
        \item $f_{156}(x) = (x+1)^n e^{-x}$
        \item $f_{157}(x) = \ln| \frac{e^x - 1}{e^x + 1} |$
        \item $f_{158}(x) = e^{x^2} \ln(x^2+1)$
        \item $f_{159}(x) = \frac{x e^x}{x+1}$
        \item $f_{160}(x) = \ln(2e^x + 3)$

        \item $f_{161}(x) = e^{\frac{x-1}{x+1}}$
        \item $f_{162}(x) = \frac{\ln(e^x + x)}{x}$
        \item $f_{163}(x) = (e^x + x)^2$
        \item $f_{164}(x) = \sqrt{x^2+1} e^{-x}$
        \item $f_{165}(x) = \ln(1 + x e^x)$
        \item $f_{166}(x) = \frac{e^{3x}}{x^3+1}$
        \item $f_{167}(x) = e^{x} (x - \ln x)$
        \item $f_{168}(x) = \frac{1}{e^x + e^{-x}}$
        \item $f_{169}(x) = \ln(x^2 e^{-x})$
        \item $f_{170}(x) = (x^3 - x) e^{1/x}$
        \item $f_{171}(x) = \frac{e^x}{\sqrt{x^2+x+1}}$
        \item $f_{172}(x) = \ln| \sin(e^x) |$
        \item $f_{173}(x) = e^{x} \tan(e^x)$
        \item $f_{174}(x) = \frac{x^2 - e^x}{x^2 + e^x}$
        \item $f_{175}(x) = \ln(e^x + e^{-x})$
        \item $f_{176}(x) = e^{\sqrt{\ln x}}$
        \item $f_{177}(x) = \frac{\ln(x)}{e^x - 1}$
        \item $f_{178}(x) = (x - e^x)^n$
        \item $f_{179}(x) = \sqrt{\frac{e^x}{e^x+1}}$
        \item $f_{180}(x) = \ln(x^2 + e^{2x})$

        \item $f_{181}(x) = e^{x^2 - x} \ln(x)$
        \item $f_{182}(x) = \frac{x e^x}{1 + e^{2x}}$
        \item $f_{183}(x) = \ln(x + e^{-x})$
        \item $f_{184}(x) = (1 - x^2) e^{-x^2}$
        \item $f_{185}(x) = \frac{e^x - \ln x}{e^x + \ln x}$
        \item $f_{186}(x) = e^{\sin(x) + \cos(x)}$
        \item $f_{187}(x) = \sqrt{e^x + x}$
        \item $f_{188}(x) = \ln(1 + \sqrt{e^x})$
        \item $f_{189}(x) = \frac{x^n}{e^{nx}}$
        \item $f_{190}(x) = (x+1) e^{1/(x+1)}$
        \item $f_{191}(x) = \ln| \frac{1}{1 - e^x} |$
        \item $f_{192}(x) = e^x \ln(x^2+1)$
        \item $f_{193}(x) = \frac{e^x + e^{-x}}{e^x - e^{-x}}$
        \item $f_{194}(x) = \ln(1 + x^2 e^x)$
        \item $f_{195}(x) = e^{x} (\cos x - \sin x)$
        \item $f_{196}(x) = \frac{\sqrt{e^x}}{x + 1}$
        \item $f_{197}(x) = \ln(e^x + e^{2x})$
        \item $f_{198}(x) = (x^2+1) e^{\sqrt{x}}$
        \item $f_{199}(x) = \frac{e^x}{x \ln^2(x)}$
        \item $f_{200}(x) = \ln(\ln(e^x+1))$

\end{enumerate}
\end{multicols}

\newpage
\section*{Exemples de Corrections Détaillées}

\subsection*{2. $f_2(x) = xe^x$}
\begin{itemize}
    \item \textbf{Limites :} $\lim_{x \to -\infty} f_2(x) = 0$ (croissance comparée). $\lim_{x \to +\infty} f_2(x) = +\infty$.
    \item \textbf{Dérivée :} $f_2'(x) = 1 \cdot e^x + x \cdot e^x = e^x(x+1)$.
\end{itemize}

\subsection*{11. $f_{11}(x) = e^{1/x}$}
\begin{itemize}
    \item \textbf{Limites :} $\lim_{x \to 0^+} f_{11}(x) = e^{+\infty} = +\infty$. $\lim_{x \to 0^-} f_{11}(x) = e^{-\infty} = 0$. $\lim_{x \to \pm\infty} f_{11}(x) = e^0 = 1$.
    \item \textbf{Dérivée :} $f_{11}'(x) = -\frac{1}{x^2}e^{1/x}$.
\end{itemize}

\subsection*{19. $f_{19}(x) = \ln(e^x + 1)$}
\begin{itemize}
    \item \textbf{Limites :} $\lim_{x \to -\infty} \ln(0+1) = 0$. $\lim_{x \to +\infty} \ln(e^x(1 + e^{-x})) = \ln(e^x) = +\infty$.
    \item \textbf{Dérivée :} $f_{19}'(x) = \frac{(e^x+1)'}{e^x+1} = \frac{e^x}{e^x+1}$.
\end{itemize}

\subsection*{100. $f_{100}(x) = xe^x - e^x + 1$}
\begin{itemize}
    \item \textbf{Dérivée :} $f_{100}'(x) = (e^x + xe^x) - e^x = xe^x$.
    \item \textbf{Note :} C'est la méthode classique pour trouver la primitive de $xe^x$.
\end{itemize}

\end{document}