# Test: Differentiation II - Ambitious

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Question 1:   Find a derivative of the function  $y={e}^{\left({x}^{7}+3{x}^{2}\right)}$
$y\text{'}=\left({x}^{7}+3{x}^{2}\right){e}^{\left({x}^{7}+3{x}^{2}\right)}$
$y\text{'}=\left(7{x}^{6}+6x\right){e}^{\left({x}^{7}+3{x}^{2}\right)}$
$y\text{'}=\left({x}^{6}+3x\right){e}^{\left({x}^{7}+3{x}^{2}\right)}$
$y\text{'}={e}^{\left({x}^{7}+3{x}^{2}\right)}$
Question 2:   Find a derivative of the function  $y={\left(x-2\right)}^{2}$
$y\text{'}=6\left(x-2\right)$
$y\text{'}=6\left(3x-2\right)$
$y\text{'}=3x-2$
$y\text{'}=2\left(3x-2\right)$
Question 3:   Find a derivative of the function  $y=arctg\left(3x\right)$
$y\text{'}=\frac{1}{1+9{x}^{2}}$
$y\text{'}=\frac{3}{1+9{x}^{2}}$
$y\text{'}=-\frac{3}{1+9{x}^{2}}$
$y\text{'}=\frac{3}{1+3x}$
Question 4:   Find a derivative of the function  $y=tg\left({x}^{5}-x\right)$
$y\text{'}=\frac{1}{{\mathrm{cos}}^{2}\left({x}^{5}-x\right)}$
$y\text{'}=\frac{5{x}^{4}-1}{{\mathrm{cos}}^{2}\left({x}^{5}-x\right)}$
$y\text{'}=\frac{5{x}^{4}-1}{{\mathrm{cos}}^{2}\left({x}^{5}\right)}$
$y\text{'}=\frac{5{x}^{4}}{{\mathrm{cos}}^{2}\left({x}^{5}-x\right)}$
Question 5:   The Newton's first law of motion is expressed by the formula  $s\left(t\right)={t}^{3}-7$ ,  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  ${t}_{0}=2$
$v=5$
$v=12$
$v=10$
$v=1$
Question 6:   The Newton's first law of motion is expressed by the formula  $s\left(t\right)={t}^{2}+4t$ ,  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  ${t}_{0}=1$ .
$v=6$
$v=9$
$v=5$
$v=4$
Question 7:   The Newton's first law of motion is expressed by the formula  $s\left(t\right)=\text{​}{t}^{4}-2{t}^{2}$ ,  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  ${t}_{0}=3$ .
$v=27$
$v=81$
$v=6$
$v=108$
Question 8:   In which point $f\text{'}\left({x}_{0}\right)>0$ ?

${x}_{0}=6$
${x}_{0}=10$
${x}_{0}=4$
${x}_{0}=8$
Question 9:   In which point $f\text{'}\left({x}_{0}\right)=0$ ?

${x}_{0}=0$
${x}_{0}=-2$
${x}_{0}=1$
${x}_{0}=-1$
Question 10:   In which point $f\text{'}\left({x}_{0}\right)<0$ ?

${x}_{0}=1$
${x}_{0}=0$
${x}_{0}=3$
${x}_{0}=-2$