9000070810
Level:
A
Differentiate the following function.
\[
f(x)=\log _{5}12
\]
\(f'(x) = 0;\ x\in \mathbb{R}\)
\(f'(x) = \frac{1}
{\ln 12};\ x\in \mathbb{R}\)
\(f'(x) = \frac{1}
{12\ln 5};\ x\in \mathbb{R}\)
\(f'(x) = 1;\ x\in \mathbb{R}\)
Derivative
9000070806
Level:
A
Differentiate the following function.
\[
f(x) = \frac{\pi }
{x} +\ln 2
\]
\(f'(x) = - \frac{\pi }{x^{2}} ;\ x\in \mathbb{R}\setminus \{0\}\)
\(f'(x) = 0;\ x\in \mathbb{R}\setminus \{0\}\)
\(f'(x) =\pi ;\ x\in \mathbb{R}\setminus \{0\}\)
\(f'(x) = \frac{\pi } {x^{2}} ;\ x\in \mathbb{R}\setminus \{0\}\)
9000070802
Level:
A
Differentiate the following function.
\[
f(x) = 3 - 2\cos x
\]
\(f'(x) = 2\sin x;\ x\in \mathbb{R}\)
\(f'(x) = 3 + 2\sin x;\ x\in \mathbb{R}\)
\(f'(x) = 3 - 2\sin x;\ x\in \mathbb{R}\)
\(f'(x) = 2\cos x;\ x\in \mathbb{R}\)
9000070701
Level:
B
Differentiate the following function.
\[
f(x)= (2x - 5)^{-6}
\]
\(f^{\prime}(x) = - \frac{12}
{(2x-5)^{7}} ;\ x\in \mathbb{R}\setminus \left \{\frac{5}
{2}\right \}\)
\(f^{\prime}(x) = - \frac{12}
{(2x-5)^{7}} ;\ x\in \mathbb{R}\)
\(f^{\prime}(x) = - \frac{12}
{(2x-5)^{5}} ;\ x\in \mathbb{R}\setminus \left \{\frac{5}
{2}\right \}\)
\(f^{\prime}(x) = - \frac{12}
{(2x-5)^{5}} ;\ x\in \left (\frac{5}
{2};\infty \right )\)
9000070705
Level:
B
Differentiate the following function.
\[
f(x) =\ln (2x^{2} + 5x)
\]
\(f^{\prime}(x) = \frac{4x+5}
{2x^{2}+5x};\ x\in \left (-\infty ;-\frac{5}
{2}\right )\cup \left (0;\infty \right )\)
\(f^{\prime}(x) = \frac{4x+5}
{2x^{2}+5x};\ x\in \mathbb{R}\setminus \left \{-\frac{5}
{2};0\right \}\)
\(f^{\prime}(x) = \frac{1}
{2x^{2}+5x};\ x\in \left (-\infty ;-\frac{5}
{2}\right )\cup \left (0;\infty \right )\)
\(f^{\prime}(x) = \frac{1}
{2x^{2}+5x};\ x\in \mathbb{R}\setminus \left \{-\frac{5}
{2};0\right \}\)
9000070702
Level:
B
Differentiate the following function.
\[
f(x) = (x^{2} - 3x + 2)^{\frac{1}
{2} }
\]
\(f^{\prime}(x) = \frac{2x-3}
{2\sqrt{x^{2 } -3x+2}};\ x\in \mathbb{R}\setminus \left [ 1;2\right ] \)
\(f^{\prime}(x) = \frac{2x-3}
{2\sqrt{x^{2 } -3x+2}};\ x\in \mathbb{R}\setminus \left (1;2\right )\)
\(f^{\prime}(x) = (4x - 6)\sqrt{x^{2 } - 3x + 2};\ x\in \mathbb{R}\setminus \left [ 1;2\right ] \)
\(f^{\prime}(x) = (4x - 6)\sqrt{x^{2 } - 3x + 2};\ x\in \mathbb{R}\setminus \left (1;2\right )\)
9000070708
Level:
B
Differentiate the following function.
\[
f(x) =\ln \left (\frac{1 + x}
{1 - x}\right )
\]
\(f^{\prime}(x) = \frac{2}
{1-x^{2}} ;\ x\in \left (-1;1\right )\)
\(f^{\prime}(x) = \frac{2}
{1-x^{2}} ;\ x\in \mathbb{R}\setminus \left \{-1;1\right \}\)
\(f^{\prime}(x) = \frac{1-x}
{1+x};\ x\in \left (-1;1\right )\)
\(f^{\prime}(x) = \frac{1-x}
{1+x};\ x\in \mathbb{R}\setminus \left \{-1;1\right \}\)
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.
\[
f(x) = \frac{1}
{\cos x + 3x^{2}}
\]
\(f^{\prime}(x) = \frac{\sin x-6x}
{(3x^{2}+\cos x)^{2}} ;\ x\in \mathbb{R}\)
\(f^{\prime}(x) = \frac{6x-\sin x}
{(3x^{2}+\cos x)^{2}} ;\ x\in \mathbb{R}\)
\(f^{\prime}(x) = \frac{\sin x-6x}
{3x^{2}+\cos x};\ x\in \mathbb{R}\)
\(f^{\prime}(x) = \frac{6x-\sin x}
{3x^{2}+\cos x};\ x\in \mathbb{R}\)
9000070706
Level:
B
Differentiate the following function.
\[
f(x) = \sqrt{x^{2 } + 3x}
\]
\(f^{\prime}(x) = \frac{2x+3}
{2\sqrt{x^{2 } +3x}};\ x\in \left (-\infty ;-3\right )\cup \left (0;\infty \right )\)
\(f^{\prime}(x) = \frac{2x+3}
{2\sqrt{x^{2 } +3x}};\ x\in \left (-\infty ;-3\right ] \cup \left [ 0;\infty \right )\)
\(f^{\prime}(x) = \frac{2x+3}
{\sqrt{x^{2 } +3x}};\ x\in \left (-\infty ;-3\right )\cup \left (0;\infty \right )\)
\(f^{\prime}(x) = \frac{\sqrt{x^{2 } +3x}}
{2x+3} ;\ x\in \left (-\infty ;-3\right ] \cup \left [ 0;\infty \right )\)