Derivatives of Functions

2010002008

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

2010002009

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

9000063101

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

9000063104

Level: 
B
Differentiate the following function. \[ f(x)= \frac{\sin x} {\sin x -\cos x} \]
\(f'(x) = \frac{-1} {(\sin x-\cos x)^{2}} ,\ x\neq \frac{\pi }{4} + k\pi ,k\in \mathbb{Z}\)
\(f'(x) = \frac{\sin ^{2}x-\cos ^{2}x} {(\sin x-\cos x)^{2}} ,\ x\neq \frac{\pi }{4} + k\pi ,k\in \mathbb{Z}\)
\(f'(x) = \frac{\sin x(\cos x+1)} {(\sin x-\cos x)^{2}} ,\ x\neq \frac{\pi }{4} + k\pi ,k\in \mathbb{Z}\)
\(f'(x) = \frac{\cos ^{2}x-\sin ^{2}x} {(\sin x-\cos x)^{2}} ,\ x\neq \frac{\pi }{4} + k\pi ,k\in \mathbb{Z}\)

9000063105

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

9000063107

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

9000063108

Level: 
B
Differentiate the following function. \[ f(x) = x^{5}\mathrm{e}^{x} \]
\(f'(x) = x^{4}\mathrm{e}^{x}(5 + x),\ x\in \mathbb{R}\)
\(f'(x) = 5x^{4}\mathrm{e}^{x},\ x\in \mathbb{R}\)
\(f'(x) = x^{4}\mathrm{e}^{x}(x - 5),\ x\in \mathbb{R}\)
\(f'(x) = x^{4}\mathrm{e}^{x}(5 + x^{2}),\ x\in \mathbb{R}\)