Generating...                               quiz12_n0

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

    $\displaystyle e^6$ $\displaystyle 6\,e^6$ $\displaystyle 9\,e^6$ $\displaystyle 3\,e^6$

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

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

  3. Suppose that $g( 5 ) = 6 $ and $g'( 5 ) = 3 $. Then find the derivative $f'( 5 )$ for $f(x) = \displaystyle {{1}\over{\left(g\left(x\right)+3\right)^2}} $ .

    $\displaystyle {{1}\over{81}}$ $\displaystyle -{{2}\over{729}}$ $\displaystyle -{{4}\over{243}}$ $\displaystyle -{{2}\over{243}}$

  4. Find the derivative $f'(x)$ for $f(x) = \displaystyle \sqrt[ 4 ]{ x^2+3\,x+1 }$ .

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

  5. Find the derivative $f'(x)$ for $f(x) =\displaystyle \sqrt{x}\,e^{x^2} $ .

    $f'(x) =\displaystyle 2\,x^{{{3}\over{2}}}\,e^{x^2} $ $f'(x) =\displaystyle x^2\,\left(\sqrt{x}\,e^{x}+{{e^{x}}\over{2\,\sqrt{x}}}\right) $ $f'(x) =\displaystyle \sqrt{x}\,e^{x^2} $ $f'(x) =\displaystyle 2\,x^{{{3}\over{2}}}\,e^{x^2}+{{e^{x^2}}\over{2\,\sqrt{x}}} $ $f'(x) =\displaystyle \sqrt{x}\,e^{x}+{{e^{x}}\over{2\,\sqrt{x}}} $

  6. Find the derivative $f'(x)$ for $f(x) =\displaystyle \cos \cos x $ .

    $f'(x) =\displaystyle \cos x\,\sin x\,\sin \cos x-\sin x\,\cos \cos x $ $f'(x) =\displaystyle \sin x\,\sin \cos x $ $f'(x) =\displaystyle -\sin x\,\cos \cos x $ $f'(x) =\displaystyle \cos x\,\cos \cos x $ $f'(x) =\displaystyle -\sin \cos x $

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

    $f'(x) =\displaystyle 6\,\left(x^2+1\right)^2\,\left(\left(x^2+1\right)^2+1\right)^2 $ $f'(x) =\displaystyle 6\,x\,\left(\left(x^2+1\right)^2+1\right)^3 $ $f'(x) =\displaystyle 6\,\left(x^2+1\right)\,\left(\left(x^2+1\right)^2+1\right)^2 $ $f'(x) =\displaystyle 4\,x\,\left(x^2+1\right)\,\left(\left(x^2+1\right)^2+1\right)^3 $ $f'(x) =\displaystyle 2\,x\,\left(\left(x^2+1\right)^2+1\right)^3 $ $f'(x) =\displaystyle 12\,x\,\left(x^2+1\right)\,\left(\left(x^2+1\right)^2+1\right)^2 $

  8. Find the derivative $f'(x)$ for $f(x) =\displaystyle {{1}\over{\sqrt{x+{{1}\over{\sqrt{x}}}}}} $ .

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

  9. Find the derivative $f'(x)$ for $f(x) =\displaystyle e^x \cot \left({{1}\over{\sqrt{x}}}\right) $ .

    $f'(x) =\displaystyle {{\left(\csc \left({{1}\over{\sqrt{x}}}\right)\right)^2\,e... ... }\over{2\,x^{{{3}\over{2}}}}}+\cot \left({{1}\over{\sqrt{x}}}\right) \,e^{x} $ $f'(x) =\displaystyle {{\left(\sec \left({{1}\over{\sqrt{x}}}\right)\right)^2\,e... ... }\over{2\,x^{{{3}\over{2}}}}}-\tan \left({{1}\over{\sqrt{x}}}\right) \,e^{x} $ $f'(x) =\displaystyle \tan \left({{1}\over{\sqrt{x}}}\right)\,e^{x}-{{\left(\sec... ...t({{ 1}\over{\sqrt{x}}}\right)\right)^2\,e^{x}}\over{2\,x^{{{3}\over{2}}} }} $ $f'(x) =\displaystyle -{{\left(\csc \left({{1}\over{\sqrt{x}}}\right)\right)^2\,... ... }\over{2\,x^{{{3}\over{2}}}}}-\cot \left({{1}\over{\sqrt{x}}}\right) \,e^{x} $

  10. Find the derivative $f'(x)$ for $f(x) =\displaystyle {{1}\over{5^{x}}} $ .

    $f'(x) =\displaystyle {{1}\over{5^{x}}} $ $f'(x) =\displaystyle -\ln 5\,e^{x} $ $f'(x) =\displaystyle -{{\ln 5}\over{5^{x}}} $ $f'(x) =\displaystyle x\,5^{1-x} $ $f'(x) =\displaystyle {{\ln 5}\over{5^{x}}} $


Department of Mathematics
Last modified: 2025-10-30