Derivative

1003230203

Level: 
B
Given the function \( f(x)=\frac{\sqrt x}{\ln ⁡x} \), find the set of all \( x \), \( x\in\mathbb{R} \), for which \( f'(x)=0 \).
\( \left\{ \mathrm{e}^2 \right\} \)
\( \{ \mathrm{e} \} \)
\( \left\{ \sqrt{\mathrm{e}} \right\} \)
\( \left\{ \frac1{\mathrm{e}};\mathrm{e} \right\} \)
\( \{ 2 \} \)
\( \left\{ 1;\mathrm{e}^2 \right\} \)

1003230204

Level: 
B
Which of the statements A, B, C given bellow are correct? \[ \begin{array}{l} \text{A: }\left(\frac1{x^3}\cdot\cos ⁡x\right)' =-\frac{\cos x+\sin ⁡x}{x^4},\ x\in\mathbb{R}\setminus\{0\} \\ \text{B: }\bigl(\left(1-x^3\right)\cdot\ln x \bigr)'=-3x^2\ln x+\frac1x - x^2,\ x\in\mathbb{R}^+ \\ \text{C: } \left(5^x\cdot\sqrt[5]x\right)'=5^{x-1}\sqrt[5]x\left(5\ln⁡5+\frac1x\right),\ x\in\mathbb{R}\setminus\{0\} \end{array} \] The only correct statements are:
B, C
A, C
A, B
B
A
C

1003230205

Level: 
B
Which of the statements A, B, C given bellow are correct? \[ \begin{array}{l} \text{A: } \left(\frac{2x-1}{2-x}\right)'=\frac{5-4x}{(2-x)^2},\ x\neq2 \\ \text{B: } \left(\frac{\mathrm{e}^x-1}{x}\right)'=\frac{\mathrm{e}^x(x-1)-1}{x^2},\ x\neq0 \\ \text{C: } \left(\frac{\cos⁡ x}{1-\sin ⁡x}\right)'=\frac1{1-\sin ⁡x},\ x\neq\frac{\pi}2+2k\pi,\ k\in\mathbb{Z} \end{array}\] The only correct statements are:
C
A, C
A, B
B
A
B, C

1003230206

Level: 
B
Which of the statements A, B, C, D given bellow are incorrect? \[ \begin{array}{l} \text{A: }\left(\ln\frac x2\right)'=\frac1x,\ x\in\mathbb{R}^+ \\ \text{B: }\left(5\sin⁡3x\right)'=5\cos⁡3x \\ \text{C: }\left(\frac1{\left(x^3-1\right)^2}\right)'=\frac{-6x^2}{\left(x^3-1\right)^3},\ x\in\mathbb{R}\setminus\{1\} \\ \text{D: }\left(\ln⁡(1+\cos⁡ x ) \right)'=\frac1{1-\sin ⁡x},\ x\neq\frac{\pi}2+2k\pi,\ k\in\mathbb{Z} \end{array}\] The only incorrect statements are:
B, D
A, B, D
B, C
B
A, C
A, C, D

2010002003

Level: 
B
Differentiate the following function. \[ f(x) = \mathrm{e}^{x}x^{4} \]
\(f'(x) = \mathrm{e}^{x}x^{3}(x+4),\ x\in \mathbb{R}\)
\(f'(x) = \mathrm{e}^{x}4x^{3},\ x\in \mathbb{R}\)
\(f'(x) = \mathrm{e}^{x}x^{3}(x - 4),\ x\in \mathbb{R}\)
\(f'(x) = \mathrm{e}^{x}x^{3}(x + x\mathrm{e}^{x}),\ x\in \mathbb{R}\)

2010002004

Level: 
B
Differentiate the following function. \[ f(x) =\cos x(1 -\mathop{\mathrm{cotg}}\nolimits x) \]
\(f'(x) =-\sin x +\cos x + \frac{\cos x} {\sin ^{2}x},\ x\in \mathbb{R}\setminus\{k\pi; k\in \mathbb{Z}\}\)
\(f'(x) = \frac{\cos x} {\sin ^{2}x},\ x\in \mathbb{R}\setminus\{k\pi; k\in \mathbb{Z}\}\)
\(f'(x) =-\sin x +\cos x,\ x\in \mathbb{R}\setminus\{k\pi; k\in \mathbb{Z}\}\)
\(f'(x) =-\sin x +2\cos x,\ x\in \mathbb{R}\setminus\{k\pi; k\in \mathbb{Z}\}\)

2010002005

Level: 
B
Differentiate the following function. \[ f(x)=\cos \left(3-2x^{2} \right) \]
\(f'(x) = 4x\sin \left(3-2x^{2} \right),\ x\in \mathbb{R}\)
\(f'(x) = -4x\sin x,\ x\in \mathbb{R}\)
\(f'(x) = -\sin \left(4x\right),\ x\in \mathbb{R}\)
\(f'(x) = \cos\left(4x+1\right),\ x\in \mathbb{R}\)

2010002006

Level: 
B
Differentiate the following function. \[ f(x) = \frac{2-x^{2} } {4x} \]
\(f'(x) = \frac{-x^{2}-2} {4x^{2}} , \ x\in \mathbb{R} \setminus \{0\}\)
\(f'(x) = \frac{-x} {2} ,\ x\in \mathbb{R} \setminus \{0\}\)
\(f'(x) = \frac{2-x^{2}} {4x^{2}} ,\ x\in \mathbb{R} \setminus \{0\}\)
\(f'(x) = \frac{-x-2} {4x^{2}} ,\ x\in \mathbb{R} \setminus \{0\}\)

2010002007

Level: 
B
Differentiate the following function. \[ f(x) = \frac{\sqrt{x} +2} {2-\sqrt{x} } \]
\(f'(x) = \frac{2} {\sqrt{x}}\frac{1}{(2-\sqrt{x})^{2}}, \ x \in (0 ;4)\cup (4;\infty) \)
\(f'(x) = \frac{\sqrt{x}} {2(2-\sqrt{x})^2}, \ x \in (0 ;4) \cup (4;\infty) \)
\(f'(x) = \frac{1} {(2-\sqrt{x})^2}, \ x \in (0 ;4) \cup (4;\infty) \)
\(f'(x) = \frac{1} {x(2-\sqrt{x})^2}, \ x \in (0 ;4) \cup (4;\infty) \)

2010002008

Level: 
B
Differentiate the following function. \[ f(x) =\ln \left(3x^{2} - 5x \right) \]
\(f^{\prime}(x) = \frac{6x-5} {3x^{2}-5x};\ x\in \left (-\infty ;0\right )\cup \left (\frac{5}{3};\infty \right )\)
\(f^{\prime}(x) = \frac{6x-5} {3x^{2}-5x};\ x\in \mathbb{R}\setminus \left \{0;\frac{5} {3}\right \}\)
\(f^{\prime}(x) = \frac{1} {3x^{2}-5x};\ x\in \left (-\infty ;0\right )\cup \left (\frac{5}{3};\infty \right )\)
\(f^{\prime}(x) = \frac{1} {3x^{2}-5x};\ x\in \mathbb{R}\setminus \left \{0;\frac{5} {3}\right \}\)