Derivatives of Functions

9000070806

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

9000070807

Level: 
B
Differentiate the following function. \[ f(x) = \frac{x^{4} + 3} {x^{2}} + x^{3} \]
\(f'(x) = 3x^{2} + 2x - \frac{6} {x^{3}} ,\ x\in \mathbb{R}\setminus \{0\}\)
\(f'(x) = 6x^{2} - 2x - \frac{6} {x^{3}} ,\ x\in \mathbb{R}\setminus \{0\}\)
\(f'(x) = 3x^{2} + 2x + \frac{6} {x^{3}} ,\ x\in \mathbb{R}\setminus \{0\}\)
\(f'(x) = 6x^{2} - 2x + \frac{6} {x^{3}} ,\ x\in \mathbb{R}\setminus \{0\}\)

9000070808

Level: 
B
Differentiate the following function. \[ f(x)= \frac{x} {x + 1} \]
\(f'(x) = \frac{1} {(x+1)^{2}} ,\ x\in \mathbb{R}\setminus \{ - 1\}\)
\(f'(x) = - \frac{1} {(x+1)^{2}} ,\ x\in \mathbb{R}\setminus \{ - 1\}\)
\(f'(x) = \frac{x} {(x+1)^{2}} ,\ x\in \mathbb{R}\setminus \{ - 1\}\)
\(f'(x) = - \frac{x} {(x+1)^{2}} ,\ x\in \mathbb{R}\setminus \{ - 1\}\)

9000070701

Level: 
B
Differentiate the following function. \[ f(x)= (2x - 5)^{-6} \]
\(f^{\prime}(x) = - \frac{12} {(2x-5)^{7}} ,\ x\in \mathbb{R}\setminus \left \{\frac{5} {2}\right \}\)
\(f^{\prime}(x) = - \frac{12} {(2x-5)^{7}} ,\ x\in \mathbb{R}\)
\(f^{\prime}(x) = - \frac{12} {(2x-5)^{5}} ,\ x\in \mathbb{R}\setminus \left \{\frac{5} {2}\right \}\)
\(f^{\prime}(x) = - \frac{12} {(2x-5)^{5}} ,\ x\in \left (\frac{5} {2},\infty \right )\)

9000070702

Level: 
B
Differentiate the following function. \[ f(x) = (x^{2} - 3x + 2)^{\frac{1} {2} } \]
\(f^{\prime}(x) = \frac{2x-3} {2\sqrt{x^{2 } -3x+2}},\ x\in \mathbb{R}\setminus \left [ 1,2\right ] \)
\(f^{\prime}(x) = \frac{2x-3} {2\sqrt{x^{2 } -3x+2}},\ x\in \mathbb{R}\setminus \left (1,2\right )\)
\(f^{\prime}(x) = (4x - 6)\sqrt{x^{2 } - 3x + 2},\ x\in \mathbb{R}\setminus \left [ 1,2\right ] \)
\(f^{\prime}(x) = (4x - 6)\sqrt{x^{2 } - 3x + 2},\ x\in \mathbb{R}\setminus \left (1,2\right )\)

9000070703

Level: 
B
Differentiate the following function. \[ f(x)= \sqrt{\sin x -\cos x} \]
\(f^{\prime}(x) = \frac{\sin x+\cos x} {2\sqrt{\sin x-\cos x}},\ x\in \left ( \frac{\pi }{4} + 2k\pi , \frac{5\pi } {4} + 2k\pi \right ),\ k\in \mathbb{Z}\)
\(f^{\prime}(x) = \frac{\sin x+\cos x} {2\sqrt{\sin x-\cos x}},\ x\in \left [ \frac{\pi }{4} + 2k\pi , \frac{5\pi } {4} + 2k\pi \right ] ,\ k\in \mathbb{Z}\)
\(f^{\prime}(x) = \frac{\sin x-\cos x} {2\sqrt{\sin x-\cos x}},\ x\in \left [ \frac{\pi }{4} + 2k\pi , \frac{5\pi } {4} + 2k\pi \right ] ,\ k\in \mathbb{Z}\)
\(f^{\prime}(x) = \frac{\sin x-\cos x} {2\sqrt{\sin x-\cos x}},\ x\in \left ( \frac{\pi }{4} + 2k\pi , \frac{5\pi } {4} + 2k\pi \right ),\ k\in \mathbb{Z}\)

9000070704

Level: 
B
Differentiate the following function. \[ f(x) = \frac{1} {\cos x + 3x^{2}} \]
\(f^{\prime}(x) = \frac{\sin x-6x} {(3x^{2}+\cos x)^{2}} ,\ x\in \mathbb{R}\)
\(f^{\prime}(x) = \frac{6x-\sin x} {(3x^{2}+\cos x)^{2}} ,\ x\in \mathbb{R}\)
\(f^{\prime}(x) = \frac{\sin x-6x} {3x^{2}+\cos x},\ x\in \mathbb{R}\)
\(f^{\prime}(x) = \frac{6x-\sin x} {3x^{2}+\cos x},\ x\in \mathbb{R}\)