Generating...                               quiz12_n25

  1. Find the derivative $f'(x)$ for $f(x) =\displaystyle \left( {{x-1}\over{x+1}} \right)^{ {{2}\over{3}}}$ .

    $f'(x) =\displaystyle {{2\,\left(x-1\right)^{{{2}\over{3}}}}\over{\left(x+1\right)^{{{8 }\over{3}}}}} $ $f'(x) =\displaystyle {{4}\over{3\,\left(x-1\right)^{{{1}\over{3}}}\,\left(x+1\right)^{{{ 5}\over{3}}}}} $ $f'(x) =\displaystyle {{2\,\left(x+1\right)^{{{1}\over{3}}}}\over{3\,\left(x-1\right)^{{{ 1}\over{3}}}}} $ $f'(x) =\displaystyle {{2\,x^{{{2}\over{3}}}}\over{3\,\left(x-1\right)^{{{1}\ove... ...{2}\over{3}}}\,x^{{{2}\over{3}}}}\over{3\, \left(x+1\right)^{{{5}\over{3}}}}} $ $f'(x) =\displaystyle {{\left(x-1\right)^{{{5}\over{3}}}}\over{\left(x+1\right)^{{{5 }\over{3}}}}} $

  2. Suppose that $g( 3 ) = -2 $ and $g'( 3 ) = -1 $. Then find the derivative $f'( 3 )$ for $f(x) = \displaystyle e^ {- 2\,g\left(x\right) } $ .

    $\displaystyle -2\,e^4$ $\displaystyle 4\,e^4$ $\displaystyle 2\,e^4$ $\displaystyle e^4$

  3. Find the derivative $f'(x)$ for $f(x) =\displaystyle 2^{\tan x} $ .

    $f'(x) =\displaystyle \ln 2\,2^{\tan x}\,\tan x $ $f'(x) =\displaystyle 2^{\tan x}\,\left(\sec x\right)^2\,\tan x $ $f'(x) =\displaystyle \ln 2\,2^{\tan x}\,\left(\sec x\right)^2 $ $f'(x) =\displaystyle \ln 2\,2^{\tan x} $ $f'(x) =\displaystyle 2^{\tan x}\,\tan ^2x $

  4. Find the derivative $f'(x)$ for $f(x) =\displaystyle \tan \cot x $ .

    $f'(x) =\displaystyle \left(\sec \cot x\right)^2 $ $f'(x) =\displaystyle -\left(\csc x\right)^2\,\left(\sec \cot x\right)^2 $ $f'(x) =\displaystyle \cot x\,\tan \cot x $ $f'(x) =\displaystyle -\left(\csc x\right)^2\,\tan \cot x $ $f'(x) =\displaystyle \left(\sec x\right)^2\,\tan \cot x-\left(\csc x\right)^2\,\tan x\, \left(\sec \cot x\right)^2 $

  5. Suppose that $g( 2 ) = 4 $ and $g'( 2 ) = 3 $. Then find the derivative $f'( 2 )$ for $f(x) = \displaystyle \left(g\left(x\right)+1\right)^3 $ .

    75 300 125 225

  6. Find the derivative $f'(x)$ for $f(x) =\displaystyle e^{\sqrt{x}} $ .

    $f'(x) =\displaystyle \sqrt{x}\,e^{x}+{{e^{x}}\over{2\,\sqrt{x}}} $ $f'(x) =\displaystyle {{e^{x}}\over{2\,\sqrt{x}}} $ $f'(x) =\displaystyle \sqrt{x}\,e^{x} $ $f'(x) =\displaystyle {{e^{\sqrt{x}}}\over{2\,\sqrt{x}}} $ $f'(x) =\displaystyle \sqrt{x}\,e^{\sqrt{x}} $

  7. Find the derivative $f'(x)$ for $f(x) = \displaystyle \sqrt[ 3 ]{ \left(x^3+2\,x+1\right)^2 }$ .

    $f'(x) =\displaystyle \left(x^3+2\,x+1\right)^{{{5}\over{3}}} $ $f'(x) =\displaystyle {{2\,\left(3\,x^2+2\right)}\over{3\,\left(x^3+2\,x+1\right)^{{{1 }\over{3}}}}} $ $f'(x) =\displaystyle {{2}\over{3\,\left(x^3+2\,x+1\right)^{{{1}\over{3}}}}} $ $f'(x) =\displaystyle {{2\,\left(x^3+2\,x+1\right)^{{{2}\over{3}}}}\over{3\,x^{{... ...}}\,\left(3\,x^2+2\right)}\over{3\,\left( x^3+2\,x+1\right)^{{{1}\over{3}}}}} $ $f'(x) =\displaystyle \left(3\,x^2+2\right)\,\left(x^3+2\,x+1\right)^{{{2}\over{3}}} $

  8. Find the derivative $f'(x)$ for $f(x) =\displaystyle \left(\cos ^3x+2\right)^4 $ .

    $f'(x) =\displaystyle -\left(\cos ^3x+2\right)^4\,\sin x $ $f'(x) =\displaystyle -3\,\cos ^2x\,\left(\cos ^3x+2\right)^4\,\sin x $ $f'(x) =\displaystyle 12\,\cos ^2x\,\left(\cos ^3x+2\right)^3 $ $f'(x) =\displaystyle 12\,\cos ^3x\,\left(\cos ^3x+2\right)^3 $ $f'(x) =\displaystyle -12\,\cos ^2x\,\left(\cos ^3x+2\right)^3\,\sin x $ $f'(x) =\displaystyle -4\,\left(\cos ^3x+2\right)^4\,\sin x $

  9. Find the derivative $f'(x)$ for $f(x) =\displaystyle \sqrt{\tan x} $ .

    $f'(x) =\displaystyle -{{\left(\csc x\right)^2}\over{2\,\sqrt{\cot x}}} $ $f'(x) =\displaystyle {{\left(\sec x\right)^2}\over{2\,\sqrt{\tan x}}} $ $f'(x) =\displaystyle {{\left(\csc x\right)^2}\over{2\,\sqrt{\cot x}}} $ $f'(x) =\displaystyle -{{\left(\sec x\right)^2}\over{2\,\sqrt{\tan x}}} $

  10. Find the derivative $f'(x)$ for $f(x) =\displaystyle \sqrt{\sqrt{x^3+1}+1} $ .

    $f'(x) =\displaystyle {{3\,x^2\,\sqrt{\sqrt{x^3+1}+1}}\over{2\,\sqrt{x^3+1}}} $ $f'(x) =\displaystyle {{1}\over{4\,\sqrt{x^3+1}\,\sqrt{\sqrt{x^3+1}+1}}} $ $f'(x) =\displaystyle {{3\,x^2\,\sqrt{\sqrt{x^3+1}+1}}\over{2}} $ $f'(x) =\displaystyle {{\sqrt{x^3+1}}\over{4\,\sqrt{\sqrt{x^3+1}+1}}} $ $f'(x) =\displaystyle 3\,x^2\,\sqrt{\sqrt{x^3+1}+1} $ $f'(x) =\displaystyle {{3\,x^2}\over{4\,\sqrt{x^3+1}\,\sqrt{\sqrt{x^3+1}+1}}} $


Department of Mathematics
Last modified: 2026-05-20