# Test: Differentiation III - Challenging

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Question 1:   Find a derivative of the function  $y=\mathrm{cos}\left(\mathrm{ln}\left(5x\right)\right)$
$y\text{'}=\frac{-\mathrm{sin}\left(\mathrm{ln}\left(5x\right)\right)}{x}$
$y\text{'}=\frac{-\mathrm{sin}\left(\mathrm{ln}\left(5x\right)\right)}{5x}$
$y\text{'}=-\mathrm{sin}\left(\mathrm{ln}\left(5x\right)\right)$
$y\text{'}=\frac{-5\mathrm{sin}\left(\mathrm{ln}\left(5x\right)\right)}{x}$
Question 2:   Find a derivative of the function  $y={5}^{\mathrm{cos}\left(x\right)}$
$y\text{'}=-\mathrm{sin}\left(x\right){5}^{\mathrm{cos}\left(x\right)}$
$y\text{'}=\mathrm{sin}\left(x\right)\mathrm{ln}\left(5\right){5}^{\mathrm{cos}\left(x\right)}$
$y\text{'}=-\mathrm{sin}\left(x\right)\mathrm{ln}\left(5\right){5}^{\mathrm{cos}\left(x\right)}$
$y\text{'}=\mathrm{ln}\left(5\right){5}^{\mathrm{cos}\left(x\right)}$
Question 3:   Find a derivative of the function  $y={\mathrm{log}}_{2}\left({x}^{2}\right)$
$y\text{'}=\frac{2x}{\mathrm{ln}\left(2\right)}$
$y\text{'}=\frac{1}{x\mathrm{ln}\left(2\right)}$
$y\text{'}=\frac{2}{x\mathrm{ln}\left(2\right)}$
$y\text{'}=\frac{2}{x}$
Question 4:   What is the equation of the tangent to the function  $y={x}^{3}-4x-1$ ,  at the point  ${x}_{0}=2$
$y=8x-16$
$y=8x-1$
$y=8x-17$
$y=8x$
Question 5:   What is the equation of the tangent to the function  $y={x}^{2}-x+1$ ,  at the point  ${x}_{0}=0$
$y=-x+1$
$y=-x$
$y=x-1$
$y=-x+2$
Question 6:   The Newton's first law of motion is expressed by the formula  $s\left(t\right)={t}^{2}-5t$ ,  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  ${t}_{0}=3$ .
$v=3$
$v=4$
$v=2$
$v=1$
Question 7:   The Newton's first law of motion is expressed by the formula  $s\left(t\right)=\mathrm{ln}\left(t\right)+t$ ,  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  ${t}_{0}=2$ .
$v=2$
$v=1$
$v=1.5$
$v=2.5$
Question 8:   What is the amount of the critical point to the graph  $f\text{'}\left(x\right)$  in the range  $\left[-2;1\right]$ $1$
$2$
$4$
$3$
Question 9:   Find range when function  $f\left(x\right)$  is ascending, the graph represents  $f\text{'}\left(x\right)$ $\left[0;3\right]$
$\left(-\infty ;+\infty \right)$
$\left(-\infty ;3\right]$
$\left[3;+\infty \right)$
Question 10:   Find range when function  $f\left(x\right)$  is descending, the graph represents  $f\text{'}\left(x\right)$ $\left(-\infty ;-1\right]\cup \left[0;1.3\right]$
$\left(-\infty ;-1\right]$
$\left[0;1.3\right]$
$\left[-1;1.3\right]$