Table of Contents
T7
Vypočítejte derivaci funkcí:
88.
\( f (x) = \frac{1}{x} + \frac{2}{x^2} + \frac{3}{x^3} \) derivative 1/x + 2/x^2 + 3/x^3
89.
\( f(x) = \frac{2x}{1 - x^2} \) derivative 2x/(1 - x^2) pomocí podilu \( \frac{2(1 + x^2)}{(1 - x^2)^2} \)
90.
\( f (x) = \frac{1 + x − x^2}{1 − x + x^2} \) derivative (1 + x − x^2)/(1 − x + x^2)
91.
\( f (x) = \frac{x}{(1 − x)^2 (1 + x)^3} \)
derivative x/((1 − x)^2 (1 + x)^3)
92.
\( f (x) = \frac{(1 - x)^p}{(1 + x)^q} \) derivative (1 - x)^p / (1 + x)^q
93.
\( f (x) = \frac{x^p(1 - x)^q}{1 + x} \)
derivative (x^p(1 - x)^q)/(1 + x)
94.
\( f(x) = x + \sqrt{x} + \sqrt[3]{x} \) derivative x + sqrt(x) + cuberoot(x)
95.
\( f (x) = \frac{1}{x} + \frac{1}{\sqrt{x}} + \frac{1}{\sqrt[3]{x}} \)
derivative 1/x + 1/sqrt x + 1/ cuberoot x
96.
\( f (x) = \sqrt[3]{x^2} − \frac{2}{\sqrt{x}} \) derivative cuberoot(x^2) − 2/sqrt x
97.
\( f (x) = x \sqrt{1 + x^2} \)
derivative x sqrt(1 + x^2)
98.
\( f (x) = \frac{x}{\sqrt{a^2 - x^2}} \) derivative x/sqrt(a^2 - x^2)
99.
\( f (x) = \sqrt[3]{\frac{1 + x^3}{1 - x^3}} \)
derivative cuberoot (1 + x^3)/(1 - x^3)
100.
\( f (x) = \cos 2x − 2 \sin x \) derivative cos 2x − 2 sin x
101.
\( f (x) = (2 − x^2) \cos x + 2x \sin x \) (2 - x^2) cos x + 2x sin x derivative (2 - x^2) cos x + 2x sin x
102.
\( f (x) = \sin \sin \sin x \) derivative sin sin sin x
103.
\( f (x) = \frac{\sin^2 x}{\sin x^2} \)
derivative (sin^2 x) / (sin x^2)
104.
\( f (x) = \frac{\cos x}{ 2 \sin^2 x} \) derivative (cos x) / (2 sin^2 x)
105.
\( f (x) = (\sin^n x) \cos nx \) derivative (sin^n x) cos(n x)
106.
\( f (x) = \frac{1}{\cos^n x} \)
derivative 1 / cos^n x
107.
\( f (x) = \frac{\sin x − x \cos x}{\cos x + x \sin x} \) derivative (sin x - x cos x) /(cos x + x sin x)
108.
\( f (x) = \tan\frac{x}{2} − \cot\frac{x}{2} \)
derivative tan x/2 - cot x/2
109.
\( f (x) = \sin \cos^2 \tan^3 x \)
derivative sin cos^2 tan^3 x
110.
\( f (x) = e^{−x^2} \) derivative e^-x^2
111.
\( f (x) = 2^{\tan \frac{1}{x}} \)
derivative 2^tan 1/x
112.
\( f (x) = e^x ( x^2 − 2x + 2 ) \) derivative e^x ( x^2 -2x + 2 )
113.
\( f(x) = \frac{(\ln 3) \sin x + \cos x }{3^x} \) derivative ( (ln 3) sin x + cos x ) / 3^x POZOR: derivative 3^-x
114.
\( f (x) = e^x + e^{e^x} + e^{e^{e^x}} \) derivative e^x + e^e^x + e^e^e^x
115.
\( f (x) = x^{a^a} + a^{x^a} + a^{a^x} \) derivative x^a^a + a^x^x + a^a^x derivative x^a^a derivative a^x^a derivative a^a^x
116.
\( f (x) = \ln^3 x^2 \)
derivative ln^3 x^2
117.
\( f (x) = \ln \ln \ln x \) derivative ln ln ln x
118.
\( f (x) = \ln \ln^2 \ln^3 x \)
derivative ln ln^2 ln^3 x
119.
\( f (x) = \frac{1}{4} \ln \frac{x^2 - 1}{x^2 + 1} \) 1/4 ln ( (x^2 - 1) / (x^2 + 1) )
120.
\( f (x) = x(\sin \ln x − \cos \ln x) \) derivative x ( sin ln x - cos ln x )
121.
\( f(x) = \ln \tan \frac{x}{2} - (\cos x) \ln \tan x \)
derivative ln tan x/2 - (cos x) ln tan x
122.
\( f (x) = \arctan \frac{x^2}{a} \)
derivative arctan x^2 / a
123.
\( f (x) = \sqrt{x} - \arctan\sqrt{x} \)
derivative sqrt x - arctan sqrt x
124.
\( f (x) = \arccos \frac{1}{x} \) derivative arccos 1/x
125.
\( f (x) = \arcsin \sin x \) derivative arcsin sin x
126.
\( f (x) = \arccos \cos^2 x \) derivative arccos cos^2 x
127.
\( f (x) = \arctan \frac{1 + x}{1 - x} \)
derivative arctag((1 + x)/(1 - x))
128.
\( f (x) = \ln (e^x + \sqrt{1 + e^{2x}} ) \)
derivative ln (e^x + sqrt (1 + e^(2x)) )
129.
\( f (x) = \arctan ( x + \sqrt{1 + x^2} ) \)
derivative arctg ( x + sqrt (1 + x^2) )
