1103023902
Level:
A
The figure shows the derivative of one of the functions displayed below. Choose the function.
Derivatives of Functions
1103023901
Level:
A
The graph of \( f' \) is given in figure. Function \( f' \) is the derivative of a function \( f \). Choose which of the following graphs is the graph of \( f \).
9000070704
Level:
B
Differentiate the following function.
\[
f(x) = \frac{1}
{\cos x + 3x^{2}}
\]
\(f^{\prime}(x) = \frac{\sin x-6x}
{(3x^{2}+\cos x)^{2}} ,\ x\in \mathbb{R}\)
\(f^{\prime}(x) = \frac{6x-\sin x}
{(3x^{2}+\cos x)^{2}} ,\ x\in \mathbb{R}\)
\(f^{\prime}(x) = \frac{\sin x-6x}
{3x^{2}+\cos x},\ x\in \mathbb{R}\)
\(f^{\prime}(x) = \frac{6x-\sin x}
{3x^{2}+\cos x},\ x\in \mathbb{R}\)
9000070705
Level:
B
Differentiate the following function.
\[
f(x) =\ln (2x^{2} + 5x)
\]
\(f^{\prime}(x) = \frac{4x+5}
{2x^{2}+5x},\ x\in \left (-\infty ,-\frac{5}
{2}\right )\cup \left (0,\infty \right )\)
\(f^{\prime}(x) = \frac{4x+5}
{2x^{2}+5x},\ x\in \mathbb{R}\setminus \left \{-\frac{5}
{2},0\right \}\)
\(f^{\prime}(x) = \frac{1}
{2x^{2}+5x},\ x\in \left (-\infty ,-\frac{5}
{2}\right )\cup \left (0,\infty \right )\)
\(f^{\prime}(x) = \frac{1}
{2x^{2}+5x},\ x\in \mathbb{R}\setminus \left \{-\frac{5}
{2},0\right \}\)
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.
\[
f(x) =\ln \left (\frac{1 + x}
{1 - x}\right )
\]
\(f^{\prime}(x) = \frac{2}
{1-x^{2}} ,\ x\in \left (-1,1\right )\)
\(f^{\prime}(x) = \frac{2}
{1-x^{2}} ,\ x\in \mathbb{R}\setminus \left \{-1,1\right \}\)
\(f^{\prime}(x) = \frac{1-x}
{1+x},\ x\in \left (-1,1\right )\)
\(f^{\prime}(x) = \frac{1-x}
{1+x},\ x\in \mathbb{R}\setminus \left \{-1,1\right \}\)
9000070801
Level:
B
Differentiate the following function.
\[
f(x) = 3\sin x\cos x
\]
\(f'(x) = 3\cos (2x),\ x\in \mathbb{R}\)
\(f'(x) = 3,\ x\in \mathbb{R}\)
\(f'(x) = -3\cos x\sin x,\ x\in \mathbb{R}\)
\(f'(x) = 3(\cos x)^{2},\ x\in \mathbb{R}\)
9000070802
Level:
A
Differentiate the following function.
\[
f(x) = 3 - 2\cos x
\]
\(f'(x) = 2\sin x,\ x\in \mathbb{R}\)
\(f'(x) = 3 + 2\sin x,\ x\in \mathbb{R}\)
\(f'(x) = 3 - 2\sin x,\ x\in \mathbb{R}\)
\(f'(x) = 2\cos x,\ x\in \mathbb{R}\)
9000070803
Level:
A
Differentiate the following function.
\[
f(x) = 3x^{3} + 2x +\mathrm{e} ^{x}
\]
\(f'(x) = 9x^{2} + 2 +\mathrm{e} ^{x},\ x\in \mathbb{R}\)
\(f'(x) = 6x^{2} + 2x,\ x\in \mathbb{R}\)
\(f'(x) = 6x^{2} + 2x +\mathrm{e} ^{x},\ x\in \mathbb{R}\)
\(f'(x) = 9x^{2} + 2,\ x\in \mathbb{R}\)