# Test: Differentiation III - Ambitious

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Question 1:   Find a derivative of the function  $y=\mathrm{ln}\left({x}^{3}+7{x}^{2}\right)$
$y\text{'}=\frac{1}{{x}^{3}+7{x}^{2}}$
$y\text{'}=\frac{3{x}^{2}+14}{{x}^{3}+7{x}^{2}}$
$y\text{'}=3{x}^{2}+14$
$y\text{'}=\frac{3{x}^{2}}{{x}^{3}+7{x}^{2}}$
Question 2:   Find a derivative of the function  $y=\mathrm{sin}\left({x}^{2}\right)$
$y\text{'}=2x$
$y\text{'}=\mathrm{cos}\left({x}^{2}\right)$
$y\text{'}=-2x\mathrm{cos}\left({x}^{2}\right)$
$y\text{'}=2x\mathrm{cos}\left({x}^{2}\right)$
Question 3:   Find a derivative of the function  $y=\mathrm{arccos}\left(5x\right)$
$y\text{'}=\frac{1}{\sqrt{1-25{x}^{2}}}$
$y\text{'}=\frac{5}{\sqrt{1-25{x}^{2}}}$
$y\text{'}=-\frac{1}{\sqrt{1-25{x}^{2}}}$
$y\text{'}=-\frac{5}{\sqrt{1-25{x}^{2}}}$
Question 4:   Find a derivative of the function  $y={x}^{4}-{x}^{3}$
$y\text{'}={x}^{4}-{x}^{3}$
$y\text{'}={x}^{3}-{x}^{2}$
$y\text{'}=4{x}^{3}-{x}^{2}$
$y\text{'}=4{x}^{3}-3{x}^{2}$
Question 5:   The Newton's first law of motion is expressed by the formula  $s\left(t\right)={t}^{2}-9t$ ,  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  ${t}_{0}=15$
$v=20$
$v=21$
$v=15$
$v=9$
Question 6:   The Newton's first law of motion is expressed by the formula  $s\left(t\right)={t}^{3}+20$ ,  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  ${t}_{0}=2$
$v=20$
$v=26$
$v=32$
$v=12$
Question 7:   The Newton's first law of motion is expressed by the formula  $s\left(t\right)=\text{​}{t}^{3}-3t$ ,  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  ${t}_{0}=1$
$v=3$
$v=1$
$v=5$
$v=0$
Question 8:   In which point  $f\text{'}\left({x}_{0}\right)>0$ ? ${x}_{0}=0.5$
${x}_{0}=0$
${x}_{0}=-0.5$
${x}_{0}=2.5$
Question 9:   In which point  $f\text{'}\left({x}_{0}\right)=0$ ? ${x}_{0}=0$
${x}_{0}=1$
${x}_{0}=-1$
${x}_{0}=-2$
Question 10:   In which point  $f\text{'}\left({x}_{0}\right)<0$ ? ${x}_{0}=1$
${x}_{0}=2$
${x}_{0}=0$
${x}_{0}=-1$