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\begin{document}\everymath{\displaystyle}
	
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\boxit{4pt}{Minitest 12}\rightarrowfill\quad\boxit{3pt}{Corrigé}

\vskip +0.5cm

Graphe de $e^x$\\
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	$exp(x)$ se note aussi {\red $e^x$ } \bigskip
	
	$e^0 = {\red 1 }$\qquad $e \approx {\red 2,7 }$\bigskip
	
	$e^{-1}\approx  {\red 0,4 }$\bigskip
	
	$e^A\times e^B = {\red  e^{A+B}}$\bigskip
	
	$e^{-A} = {\red  \dfrac{1}{e^A}}$\bigskip
	
	$\dfrac{e^A}{e^B} = {\red e^{A-B} }$\bigskip
	
	
\end{minipage}

\vskip 0.05cm


Simplifier les expressions suivantes :

$\bullet\quad (e^x)^3e^{-2x} = {\red  e^{3x-2x} = e^x}$\hfil $\bullet\quad \dfrac{e^{x-1}}{e^{x+2}} = {\red e^{(x-1)-(x+2)} = e^{x-1-x-2} = e^{-3} }$\vskip 0.5cm



Calculer les dérivées des fonctions suivantes

$\bullet\quad f(x) = (x^2-2x)e^x$\hfil 	$\bullet\quad f(x) = \dfrac{e^x}{3x+2}$

\begin{minipage}{8cm}\red\doublespacing
	
Forme $u\times v$ de dérivée $u'\times v+u\times v'$

ici $u = x^2-2x$ donc $u'= 2x-2$

et $v = e^x$ donc $v'=e^x$

En reportant cela donne : 

$f'(x) = (2x-2)e^x+(x^2-2x)e^x$

donc en factorisant :

$f'(x)  =e^x\Big(2x-2+x^2-2x\Big)$

donc $f'(x) = e^x\Big(x^2-2\Big)$
\end{minipage}
\hfil
\begin{minipage}{8cm}\red\doublespacing
	
	Forme $\frac{u}{v}$ de dérivée $\frac{u'\times v-u\times v'}{v^2}$
	
	ici $u = e^x$ donc $u'= e^x$
	
	et $v = (3x+2)$ donc $v'=3$
	
	En reportant cela donne : 
	
	$f'(x) = \dfrac{e^x\times (3x+2)-e^x\times 3}{(3x+2)^2}$
	
	donc en factorisant :
	
	$f'(x)  =\dfrac{e^x\times (3x+2-3)}{(3x+2)^2}$\\
	
	donc $f'(x)  =\dfrac{e^x\times (3x-1)}{(3x+2)^2}$
\end{minipage}


\vskip 0.25cm


\'Etudier les variations de $f(x) = (2x+3)e^x$\bigskip

\begin{minipage}{9cm}\red\doublespacing
	
	Forme $u\times v$
	
	de dérivée $u'\times v+u\times v'$
	
	ici $u = 2x+3$ donc $u'= 2$
	
	et $v = e^x$ donc $v'=e^x$
	
	En reportant cela donne : $f'(x) = (2x+3)e^x+2e^x$
	
	donc en factorisant : $f'(x) = e^x\Big(2x+5\Big)$
	
	Puis $f'(x) = 0$ ssi $2x+5=0$ ou $e^x = 0$
	
	donc $x=-\frac52$ car $e^x\neq0$
	
	$x\mapsto 2x+5$ est affine croissante.
\end{minipage}
\begin{minipage}{5cm}\red\doublespacing


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\end{document} 