======T7======
Vypočítejte [[derivace|derivaci]] funkcí:

=====88.=====
\( f (x) = \frac{1}{x} + \frac{2}{x^2} + \frac{3}{x^3} \)
[[wa>derivative 1/x + 2/x^2 + 3/x^3 ]]

=====89.=====
\( f(x) = \frac{2x}{1 - x^2} \)
[[wa>derivative 2x/(1 - x^2)]]
pomocí [[derivace#podilu]]
\( \frac{2(1 + x^2)}{(1 - x^2)^2} \)

=====90.=====
\( f (x) = \frac{1 + x − x^2}{1 − x + x^2} \)
[[wa>derivative (1 + x − x^2)/(1 − x + x^2)]]

=====91.=====
DELETEME \( f (x) =  \frac{x}{(1 − x)^2 (1 + x)^3} \)
[[wa>derivative x/((1 − x)^2 (1 + x)^3) ]]

=====92.=====
\( f (x) = \frac{(1 - x)^p}{(1 + x)^q} \)
[[wa>derivative (1 - x)^p / (1 + x)^q ]]

=====93.=====
DELETEME \( f (x) = \frac{x^p(1 - x)^q}{1 + x} \)
[[wa>derivative (x^p(1 - x)^q)/(1 + x)]]

=====94.=====
\( f(x) = x + \sqrt{x} + \sqrt[3]{x} \)
[[wa> derivative x + sqrt(x) + cuberoot(x) ]]

=====95.=====
DELETEME \( f (x) = \frac{1}{x} + \frac{1}{\sqrt{x}} + \frac{1}{\sqrt[3]{x}} \)
[[wa>derivative 1/x + 1/sqrt x + 1/ cuberoot x ]]

=====96.=====
\( f (x) = \sqrt[3]{x^2} − \frac{2}{\sqrt{x}} \)
[[wa>derivative  cuberoot(x^2) − 2/sqrt x ]]

=====97.=====
DELETEME \( f (x) = x \sqrt{1 + x^2} \)
[[wa>derivative x sqrt(1 + x^2) ]]

=====98.=====
\( f (x) = \frac{x}{\sqrt{a^2 - x^2}} \)
[[wa>derivative x/sqrt(a^2 - x^2) ]]

=====99.=====
DELETEME \( f (x) = \sqrt[3]{\frac{1 + x^3}{1 - x^3}} \)
[[wa>derivative cuberoot (1 + x^3)/(1 - x^3) ]]

=====100.=====
\( f (x) = \cos 2x − 2 \sin x \)
[[wa>derivative cos 2x − 2 sin x ]]

=====101.=====
\( f (x) = (2 − x^2) \cos x + 2x \sin x \)
[[wa> (2 - x^2) cos x + 2x sin x ]]
[[wa> derivative (2 - x^2) cos x + 2x sin x ]]

=====102.=====
\( f (x) = \sin \sin \sin x \)
[[wa> derivative sin sin sin x]]

=====103.=====
DELETEME \( f (x) = \frac{\sin^2 x}{\sin x^2} \)
[[wa> derivative (sin^2 x) / (sin x^2) ]]

=====104.=====
\( f (x) = \frac{\cos x}{ 2 \sin^2 x} \)
[[wa>derivative (cos x) / (2 sin^2 x) ]]

=====105.=====
\( f (x) = (\sin^n x) \cos nx \)
[[wa>derivative (sin^n x) cos(n x) ]]

=====106.=====
DELETEME \( f (x) = \frac{1}{\cos^n x} \)
[[wa> derivative 1 / cos^n x ]]

=====107.=====
\( f (x) = \frac{\sin x − x \cos x}{\cos x + x \sin x} \)
[[wa> derivative (sin x - x cos x) /(cos x + x sin x) ]]

=====108.=====
DELETEME \( f (x) = \tan\frac{x}{2} − \cot\frac{x}{2} \)
[[wa> derivative tan x/2 - cot x/2 ]]

=====109.=====
DELETEME \( f (x) = \sin \cos^2 \tan^3 x \)
[[wa> derivative sin cos^2 tan^3 x ]]


=====110.=====
\( f (x) = e^{−x^2} \)
[[wa> derivative e^-x^2 ]]

=====111.=====
DELETEME \( f (x) = 2^{\tan \frac{1}{x}} \)
[[wa> derivative 2^tan 1/x ]]

=====112.=====
\( f (x) = e^x ( x^2 − 2x + 2 ) \)
[[wa> derivative e^x ( x^2 -2x + 2 ) ]]

=====113.=====
\( f(x) = \frac{(\ln 3) \sin x + \cos x }{3^x} \)
[[wa> derivative ( (ln 3) sin x + cos x ) / 3^x ]] POZOR: [[wa>  derivative 3^-x ]]

=====114.=====
\( f (x) = e^x + e^{e^x} + e^{e^{e^x}} \)
[[wa> derivative e^x + e^e^x + e^e^e^x ]]

=====115.=====
\( f (x) =      x^{a^a} + a^{x^a} + a^{a^x} \)
[[wa> derivative x^a^a  +  a^x^x  +  a^a^x ]]
[[wa> derivative x^a^a ]]
[[wa> derivative a^x^a ]]
[[wa> derivative a^a^x ]]

=====116.=====
DELETEME \( f (x) = \ln^3 x^2 \)
[[wa> derivative ln^3 x^2 ]]

=====117.=====
\( f (x) = \ln \ln \ln x \)
[[wa> derivative ln ln ln x ]]

=====118.=====
DELETEME \( f (x) = \ln \ln^2 \ln^3 x \)
[[wa> derivative ln ln^2 ln^3 x ]]

=====119.=====
\( f (x) = \frac{1}{4} \ln \frac{x^2 - 1}{x^2 + 1} \)
[[wa> 1/4 ln ( (x^2 - 1) / (x^2 + 1) ) ]]


=====120.=====
\( f (x) = x(\sin \ln x − \cos \ln x) \)
[[wa> derivative x ( sin ln x - cos ln x ) ]]

=====121.=====
DELETEME \( f(x) = \ln \tan \frac{x}{2} - (\cos x) \ln \tan x \)
[[wa> derivative ln tan x/2 - (cos x) ln tan x ]]

=====122.=====
DELETEME \( f (x) = \arctan \frac{x^2}{a} \)
[[wa> derivative arctan x^2 / a ]]

=====123.=====
DELETEME \( f (x) = \sqrt{x} - \arctan\sqrt{x} \)
[[wa> derivative sqrt x - arctan sqrt x ]]

=====124.=====
\( f (x) = \arccos \frac{1}{x} \)
[[wa> derivative arccos 1/x ]]

=====125.=====
\( f (x) = \arcsin \sin x \)
[[wa> derivative arcsin sin x ]]

=====126.=====
\( f (x) = \arccos \cos^2 x \)
[[wa> derivative arccos cos^2 x ]]

=====127.=====
DELETEME \( f (x) = \arctan \frac{1 + x}{1 - x} \)
[[wa> derivative arctag((1 + x)/(1 - x)) ]]

=====128.=====
DELETEME \( f (x) = \ln (e^x + \sqrt{1 + e^{2x}} ) \)
[[wa> derivative ln (e^x + sqrt (1 + e^(2x)) ) ]]

=====129.=====
DELETEME \( f (x) = \arctan ( x + \sqrt{1 + x^2} ) \)
[[wa> derivative arctg ( x + sqrt (1 + x^2) ) ]]