\subsubsection {}

Vy�et�ete spojitost a najd�te derivaci
funkc� 

1. $x^x$,\quad 

2. $\left(\frac1x\right)^{\frac1x}$,\quad

3. $\left(\sin x\right)^{\cos x}$,\quad 

4. ${\operatorname{arctg}}\left({\operatorname{tg}}^2
x\right)$,\quad 

5. $\arcsin(\sin x)$,

6. $\log\arccos x$,\quad 

7. $x\arcsin\sqrt{x\over
x+1}+{\operatorname{arctg}}\sqrt x-\sqrt x$,

8. ${\operatorname{arctg}} e^x-\log\sqrt{e^{2x}\over e^{2x}+1}$,\quad

9. ${\operatorname{arctg}}\sqrt{x^2-1}-{\log x\over\sqrt{x^2-1}}$,\quad

10. $\left({\operatorname{arctg}} x\right)^{\arcsin x}$.

\subsubsection {}
Spo�t�te limity 

a) $\lim\limits_{x\to
a}{x^x-a^a\over
x-a},\,(a>0),\quad 

b) \lim\limits_{x\to
0}\left(x+e^x\right)^{\frac1x}$.



\subsubsection {}
Vy�et�ete spojitost a derivaci (v�etn�
jednostrann�ch) funkce
$$f(x)= (x-1)^2\vert x^2-1\vert 

\mbox{pro} x\in\Bbb R\setminus\{-2,2\} $$

$$f(x) = {x^2+3x+2\over\sqrt{\vert
x^2-4\vert}} & \\ 27   \mbox{pro} 
x\in\{-2,2\}. \qquad $$

$$f(x)= (x+1)^2\vert x^2-1\vert 

\mbox{pro} x\in\Bbb R\setminus\{-2,2\} $$

$$f(x) = {x^2-x-2\over\sqrt{\vert
x^2-4\vert}} & \\ 27   \mbox{pro} 
x\in\{-2,2\}. \qquad $$


\subsubsection {}

Vy�et�ete spojitost a derivaci
(v�etn�
jednostrann�ch) funkce

$$f(x)= e^{-{1\over\ln\vert x\vert}} \mbox{pro} x\ne 0 , quad
f(x)= 1 \mbox{pro} x=0. $$

\subsubsection {}

Spo�t�te derivaci funkce
$f(x)=\sin\left({\operatorname{arccotg}}^3\left(x^{x^3}+x^{x^x}\right)\right)$.

\subsubsection {}

Spo�t�te $\lim\limits_{x\to
0}{e^{\sin x}-e^{\sin
3x}\over x}$.



\subsubsection {}
Najd�te pod grafem funkce $e^{-x}$ $x>0$ troj�heln�k
s nejv�t�� plochou.

( �e�en� : v bod� dotyku $(x,e^{-x}) $ m�me
troj�heln�k $e^{-x}(1+x)^2 /2$, extr�m $x=1$,
podle [Fraga], III.3.19 )   

