# Test: Differentiation II - Challenging

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Question 1:   Find a derivative of the function  $y=arctg\left(\mathrm{ln}\left({x}^{2}\right)\right)$
$y\text{'}=\frac{2}{1+\left(\mathrm{ln}\left({x}^{2}\right)\right)}$
$y\text{'}=\frac{1}{1+\left(\mathrm{ln}\left({x}^{2}\right)\right)}$
$y\text{'}=\frac{2}{\left(1+\left(\mathrm{ln}\left({x}^{2}\right)\right)\right)x}$
$y\text{'}=\frac{2x}{1+\left(\mathrm{ln}\left({x}^{2}\right)\right)}$
Question 2:   Find a derivative of a function   $y={\mathrm{sin}}^{2}\left(\mathrm{cos}\left(x\right)\right)$
$y\text{'}=\mathrm{sin}\left(2\mathrm{cos}\left(x\right)\right)$
$y\text{'}=2\mathrm{sin}\left(\mathrm{cos}\left(x\right)\right)$
$y\text{'}=\mathrm{sin}\left(x\right)\mathrm{sin}\left(2\mathrm{cos}\left(x\right)\right)$
$y\text{'}=-\mathrm{sin}\left(x\right)\mathrm{sin}\left(2\mathrm{cos}\left(x\right)\right)$
Question 3:   Find a derivative of a function  $y={e}^{\mathrm{cos}\left({x}^{2}\right)}$
$y\text{'}=-2x{e}^{\mathrm{cos}\left({x}^{2}\right)}\mathrm{sin}\left({x}^{2}\right)$
$y\text{'}=-{e}^{\mathrm{cos}\left({x}^{2}\right)}\mathrm{sin}\left({x}^{2}\right)$
$y\text{'}={e}^{\mathrm{cos}\left({x}^{2}\right)}$
$y\text{'}=-2x{e}^{\mathrm{cos}\left({x}^{2}\right)}$
Question 4:   What is the equation of the tangent to the function  $y={x}^{2}+7x-1$  at a point  ${x}_{0}=1$
$y=9x$
$y=9x-9$
$y=9x-2$
$y=9x+2$
Question 5:   What is the equation of the tangent to the function  $y={x}^{3}+{x}^{2}+1$  at the point  ${x}_{0}=3$
$y=33x-99$
$y=33x$
$y=33x-62$
$y=33x+37$
Question 6:   The Newton's first law of motion is expressed by the formula  $s\left(t\right)={t}^{3}-{t}^{2}$ ,  where  $t$ - time (in seconds),  $s\left(t\right)$ - deviation of a point in the moment of time  $t$  (in meters) from the initial position. Find the speed of movement at the point  ${t}_{0}=1$
$v=4$
$v=10$
$v=1$
$v=5$
Question 7:   The Newton's first law of motion is expressed by the formula  $s\left(t\right)={t}^{20}-{t}^{15}+{t}^{10}+20t$ ,  where  $t$ - time (in seconds),  $s\left(t\right)$ - deviation of a point in the moment of time  $t$  (in meters) from the initial position. Find the speed of movement at the point  ${t}_{0}=0$
$v=20$
$v=19$
$v=10$
$v=15$
Question 8:   Find the amount of the critical points to the graph  $f\text{'}\left(x\right)$  for the segment  $\left[-1.5;1\right)$

$4$
$1$
$2$
$3$
Question 9:   Find range when function  $f\left(x\right)$  is ascending, the graph represents  $f\text{'}\left(x\right)$

$\left(-\infty ;-5\right]$
$\left[-5;1\right]$
$\left(-\infty ;-5\right]\cup \left[1;+\infty \right)$
$\left[1;+\infty \right)$
Question 10:   Find range when function  $f\left(x\right)$  is descending, the graph represents  $f\text{'}\left(x\right)$

$\left[-1;1\right]$
$\left(-\infty ;1\right]$
$\left[1;+\infty \right)$
$\left(-\infty ;+\infty \right)$