# Test: Differentiation I - Challenging

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Question 1:   Find a derivative of the function  $y=\frac{1}{{\mathrm{sin}}^{2}\left(5x\right)}$
$y\text{'}=-10{\mathrm{sin}}^{-3}\left(5x\right)\mathrm{cos}\left(5x\right)$
$y\text{'}=-10\mathrm{cos}\left(5x\right)$
$y\text{'}=-2{\mathrm{sin}}^{-3}\left(5x\right)$
$y\text{'}=-10{\mathrm{sin}}^{-2}\left(5x\right)\mathrm{cos}\left(5x\right)$
Question 2:   Find a derivative of the function  $y=\mathrm{ln}\left(x\mathrm{cos}\left(x\right)\right)$
$y\text{'}=\frac{1}{x}-tg\left(x\right)$
$y\text{'}=\frac{\mathrm{cos}\left(x\right)-x}{x\mathrm{cos}\left(x\right)}$
$y\text{'}=-tg\left(x\right)$
$y\text{'}=\frac{1}{x\mathrm{cos}\left(x\right)}$
Question 3:   Find a derivative of the function  $y=tg\left({e}^{{x}^{10}}\right)$
$y\text{'}=\frac{10{x}^{9}}{{\mathrm{cos}}^{2}\left({e}^{{x}^{10}}\right)}$
$y\text{'}=\frac{{e}^{{x}^{10}}×10{x}^{9}}{{\mathrm{cos}}^{2}\left({e}^{{x}^{10}}\right)}$
$y\text{'}=\frac{{e}^{{x}^{10}}×10{x}^{9}}{{\mathrm{cos}}^{2}\left({e}^{x}\right)}$
$y\text{'}=\frac{{e}^{{x}^{10}}}{{\mathrm{cos}}^{2}\left({e}^{{x}^{10}}\right)}$
Question 4:   What is the equation of the tangent to the function  $y={x}^{2}-x+5$ ,  at the point  ${x}_{0}=2$
$y=3x-6$
$y=3x$
$y=3x-1$
$y=3x+1$
Question 5:   What is the equation of the tangent of the function  $y={x}^{4}-3{x}^{3}+x$  ,  at the point  ${x}_{0}=1$
$y=-4x$
$y=-4x+3$
$y=-4x-1$
$y=-4x+4$
Question 6:   The Newton's first law of motion is expressed by the formula  $s\left(t\right)={t}^{7}+5{t}^{2}+\mathrm{sin}\left(x\right)$ ,  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  ${t}_{0}=0$
$v=1$
$v=0$
$v=10$
$v=15$
Question 7:   The Newton's first law of motion is expressed by the formula  $s\left(t\right)={t}^{5}-7{t}^{3}$ ,  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  ${t}_{0}=2$
$v=-4$
$v=2$
$v=1$
$v=4$
Question 8:   How many critical points are in the graph  $f\text{'}\left(x\right)$  in the range  $\left(-1;2\right)$ ?

$3$
$4$
$5$
$2$
Question 9:   Find range when function  $f\left(x\right)$  is ascending, the graph represents  $f\text{'}\left(x\right)$

$\left(-\infty ;-3\right]$
$\left[-3;-2\right]$
$\left[-2;+\infty \right)$
$\left(-\infty ;-3\right]\cup \left[-2;+\infty \right)$
Question 10:   Find range when function  $f\left(x\right)$  is descending, the graph represents  $f\text{'}\left(x\right)$

$\left(-\infty ;+\infty \right)$
$\left[-4;-1.5\right]$
$\left[-4;0\right]$
$\left(-\infty ;-4\right]$