# Test: Differentiation I - Ambitious

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Question 1:   Find a derivative of the function  $y=tg\left({x}^{2}+3x\right)$
$y\text{'}=\frac{1}{{\mathrm{cos}}^{2}\left({x}^{2}\right)}$
$y\text{'}=\frac{x}{{\mathrm{cos}}^{2}\left({x}^{2}+3x\right)}$
$y\text{'}=\frac{1}{{\mathrm{cos}}^{2}\left({x}^{2}+3x\right)}$
$y\text{'}=\frac{x+3}{{\mathrm{cos}}^{2}\left({x}^{2}+3x\right)}$
Question 2:   Find a derivative of the function  $y={e}^{\left({x}^{3}+5\right)}$
$y\text{'}=\left({x}^{3}+5\right){e}^{\left({x}^{3}+5\right)}$
$y\text{'}=\left(3{x}^{2}+5\right){e}^{\left({x}^{3}+5\right)}$
$y\text{'}={e}^{\left({x}^{3}+5\right)}$
$y\text{'}=3{x}^{2}{e}^{\left({x}^{3}+5\right)}$
Question 3:   Find a derivative of the function  $y=\sqrt{\mathrm{cos}\left(x\right)}$
$y\text{'}=\frac{\mathrm{sin}\left(x\right)}{2\sqrt{\mathrm{cos}\left(x\right)}}$
$y\text{'}=\frac{-\mathrm{sin}\left(x\right)}{2\sqrt{\mathrm{cos}\left(x\right)}}$
$y\text{'}=\frac{1}{2\sqrt{\mathrm{cos}\left(x\right)}}$
$y\text{'}=\frac{1}{\sqrt{\mathrm{cos}\left(x\right)}}$
Question 4:   Find a derivative of the function  $y=\mathrm{arcsin}\left({x}^{4}+2{x}^{2}\right)$
$y\text{'}=\frac{4{x}^{3}}{\sqrt{1-{x}^{8}-4{x}^{6}+4{x}^{4}}}$
$y\text{'}=\frac{1}{\sqrt{1-{x}^{8}-4{x}^{6}+4{x}^{4}}}$
$y\text{'}=\frac{4\left({x}^{3}+x\right)}{\sqrt{1-{x}^{8}-4{x}^{6}+4{x}^{4}}}$
$y\text{'}=\frac{1}{\sqrt{1-\left({x}^{4}+2{x}^{2}\right)}}$
Question 5:   The function s is defined by:  $s\left(t\right)={t}^{2}+t$ ,  find the gradient of the function at the point  ${t}_{0}=5$
$v=1$
$v=11$
$v=23$
$v=5$
Question 6:   The function s is defined by:  $s\left(t\right)={t}^{3}-2t$ ,  find the gradient of the function at the point  ${t}_{0}=1$
$v=1$
$v=3$
$v=10$
$v=2$
Question 7:   The function s is defined by:  $s\left(t\right)=\text{​}{t}^{5}-7t$ ,  find the gradient of the function at the point  ${t}_{0}=2$
$v=66$
$v=20$
$v=5$
$v=25$
Question 8:   At which point  $f\text{'}\left({x}_{0}\right)>0$ ?

${x}_{0}=-5$
${x}_{0}=-4$
${x}_{0}=0$
${x}_{0}=-3$
Question 9:   At which point  $f\text{'}\left({x}_{0}\right)=0$ ?

${x}_{0}=5$
${x}_{0}=2$
${x}_{0}=3$
${x}_{0}=1$
Question 10:   At which point  $f\text{'}\left({x}_{0}\right)<0$ ?

${x}_{0}=8$
${x}_{0}=12$
${x}_{0}=10$
${x}_{0}=11$