Derivative

1003112011

Level: 
B
Differentiate the following function: \[ f(x)=\frac{x-5\sqrt[3]{x^2}}{\sqrt[3]x} \]
\( f'(x)=\frac{2\sqrt[3]{x^2}-5\sqrt[3] x}{3x}\text{, }x\in\mathbb{R}\setminus\{0\} \)
\( f'(x)=\frac{2\sqrt[3]{x^2}-5\sqrt[3]x }{3}\text{, }x\in\mathbb{R}\setminus\{0\} \)
\( f'(x)=\frac{2\sqrt[3]x-5\sqrt[3]{x^2}}{3x}\text{, }x\in\mathbb{R}\setminus\{0\} \)
\( f'(x)=\frac{\sqrt[3]x}{3x}-5\text{, }x\in\mathbb{R}\setminus\{0\} \)

1003112010

Level: 
B
Differentiate the following function: \[ f(x)=\frac{\sqrt{x\sqrt x}}x \]
\( f'(x)=-\frac{\sqrt[4]{x^3}}{4x^2}\text{, }x\in\mathbb{R}^+ \)
\( f'(x)=-\frac{\sqrt[4]{x^3}}{x^2}\text{, }x\in\mathbb{R}^+ \)
\( f'(x)=-\frac{\sqrt[4]x}{4x}\text{, }x\in\mathbb{R}^+ \)
\( f'(x)=\frac{\sqrt[4]x}{4x}\text{, }x\in\mathbb{R}^+ \)

1003112009

Level: 
A
Which of the statements A, B, C, D given bellow are incorrect? \[ \begin{array}{l} \text{A: }\ \left(\sqrt{x}-\cos⁡ x+\mathrm{ln}\, ⁡x\right)'=\frac1{2\sqrt x}+\sin x+\frac1x\text{, }x\in\mathbb{R}^+ \\ \text{B: }\ \left(3\sin x-3^x+x^3 \right)'=3\cos x-3^x+3x^2 \text{, }x\in\mathbb{R} \\ \text{C: }\ \left(\mathrm{cotg}\, x-\mathrm{log}_3 5\right)'=-\frac1{\sin^2 x}\text{, }x\in\mathbb{R}\setminus\bigcup\limits_{k\in\mathbb{Z}}\{k\pi\} \\ \text{D: }\ \left(3\sqrt[3]{x^2}-2\mathrm{e}^x \right)'=\frac2{\sqrt[3]x}-2\mathrm{e}^x\text{, }x\in\mathbb{R}\setminus\{0\} \end{array} \] The only incorrect statements are:
B
B, C
A, B, C
C, D
B, C, D
C

1003112008

Level: 
B
Which of the statements A, B, C, D given bellow are correct? \[ \begin{array}{l} \text{A: }\ \left(3x^{-3}-\frac5{x^2} +7\right)'=-\frac9{x^4}-\frac{10}{x^3}\text{, }x\in\mathbb{R}\setminus\{0\} \\ \text{B: }\ \left(\frac{x^3-4}{3x}\right)'=2x+\frac4{3x^2}\text{, }x\in\mathbb{R}\setminus\{0\} \\ \text{C: }\ \left(\frac{x^4-x+1}{x}\right)'=3x^2-\frac{1}{x^2}\text{, }x\in\mathbb{R}\setminus\{0\} \\ \text{D: }\ \left(\frac2{x^2}-\frac3{x^3} \right)'=-\frac4{x^3}+\frac9{x^4}\text{, }x\in\mathbb{R}\setminus\{0\} \end{array} \] The only correct statements are:
C, D
B, C
A, B
A, C, D
B, C, D
A, D

1003112007

Level: 
A
Which of the statements A, B, C, D given bellow are correct? \[ \begin{array}{l} \text{A: }\ \left(3x^3-5x^2+7x\right)'=9x^2-10x+7\text{, }x\in\mathbb{R} \\ \text{B: }\ \left(x^2+2x^{-2}-5\right)'=2x-4x^{-3}\text{, }x\in\mathbb{R}\setminus\{0\} \\ \text{C: }\ \left(5x^4-7x^3+2\pi\right)'=20x^3-21x^2+2\text{, }x\in\mathbb{R} \\ \text{D: }\ \left(x^4-\frac4{x^4}\right)'=4x^3-\frac{16}{x^3}\text{, }x\in\mathbb{R}\setminus\{0\} \end{array} \] The only correct statements are:
A, B
A, C
A, B, C
A, D
A, C, D