B

9000070706

Level: 
B
Differentiate the following function. \[ f(x) = \sqrt{x^{2 } + 3x} \]
\(f^{\prime}(x) = \frac{2x+3} {2\sqrt{x^{2 } +3x}},\ x\in \left (-\infty ,-3\right )\cup \left (0,\infty \right )\)
\(f^{\prime}(x) = \frac{2x+3} {2\sqrt{x^{2 } +3x}},\ x\in \left (-\infty ,-3\right ] \cup \left [ 0,\infty \right )\)
\(f^{\prime}(x) = \frac{2x+3} {\sqrt{x^{2 } +3x}},\ x\in \left (-\infty ,-3\right )\cup \left (0,\infty \right )\)
\(f^{\prime}(x) = \frac{\sqrt{x^{2 } +3x}} {2x+3} ,\ x\in \left (-\infty ,-3\right ] \cup \left [ 0,\infty \right )\)

9000070707

Level: 
B
Differentiate the following function. \[ f(x) = \root{5}\of{x^{2} - 7x} \]Remark: The function \(f\colon y = \root{5}\of{x}\) is defined for \(x\in \left < 0,\infty \right )\).
\(f^{\prime}(x) = \frac{2x-7} {5(x^{2}-7x)^{\frac{4} {5} }} ,\ x\in \left (-\infty ,0\right )\cup \left (7,\infty \right )\)
\(f^{\prime}(x) = \frac{2x-7} {5(x^{2}-7x)^{\frac{4} {5} }} ,\ x\in \left (-\infty ,0\right ] \cup \left [ 7,\infty \right )\)
\(f^{\prime}(x) = (2x - 7)\root{4}\of{x^{2} - 7x},\ x\in \left (-\infty ,0\right )\cup \left (7,\infty \right )\)
\(f^{\prime}(x) = (2x - 7)\root{4}\of{x^{2} - 7x},\ x\in \left (-\infty ,0\right ] \cup \left [ 7,\infty \right )\)

9000070708

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

9000065901

Level: 
B
Evaluate the following integral on the interval \((-1,+\infty)\). \[ \int \frac{1} {x + 1}\, \text{d}x \]
\(\ln |x + 1| + c,\ c\in \mathbb{R}\)
\(\ln |x| + c,\ c\in \mathbb{R}\)
\(\frac{1} {x} + c,\ c\in \mathbb{R}\)
\(-\frac{1} {2}(x + 1)^{-2} + c,\ c\in \mathbb{R}\)

9000065903

Level: 
B
Evaluate the following integral on the interval \((-6,+\infty)\). \[ \int \frac{1} {6x + 36}\, \text{d}x \]
\(\frac{1} {6}\ln |x + 6| + c,\ c\in \mathbb{R}\)
\(-\frac{1} {2}(6x + 36)^{-2} + c,\ c\in \mathbb{R}\)
\(6\ln |x + 6| + c,\ c\in \mathbb{R}\)
\(12x^{2} + 36x + c,\ c\in \mathbb{R}\)

9000065904

Level: 
B
Evaluate the following integral on the interval \((0,+\infty)\). \[ \int \frac{x^{3} + 2x} {x^{2}} \, \text{d}x \]
\(\frac{1} {2}x^{2} + 2\ln |x| + c,\ c\in \mathbb{R}\)
\(x +\ln |x| + c,\ c\in \mathbb{R}\)
\(\frac{1} {4}x^{4} + 4x^{2} +\ln |x^{2}| + c,\ c\in \mathbb{R}\)
\(2x^{2} + 2 +\ln |x^{2}| + c,\ c\in \mathbb{R}\)