Generating...                               quiz12_n7

  1. Find the derivative $f'(x)$ for $f(x) =\displaystyle \sin \tan x $ .

    $f'(x) =\displaystyle \tan x\,\sin \tan x $ $f'(x) =\displaystyle \cos x\,\sin \tan x+\left(\sec x\right)^2\,\sin x\,\cos \tan x $ $f'(x) =\displaystyle \left(\sec x\right)^2\,\sin \tan x $ $f'(x) =\displaystyle \cos \tan x $ $f'(x) =\displaystyle \left(\sec x\right)^2\,\cos \tan x $

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

    $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} $ $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 \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(\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} $

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

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

  4. 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}} $ .

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

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

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

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

    $f'(x) =\displaystyle \ln 3\,3^{\cos x}\,\cos x $ $f'(x) =\displaystyle \ln 3\,3^{\cos x} $ $f'(x) =\displaystyle -\ln 3\,3^{\cos x}\,\sin x $ $f'(x) =\displaystyle -3^{\cos x}\,\cos x\,\sin x $ $f'(x) =\displaystyle 3^{\cos x}\,\cos ^2x $

  7. Find the derivative $f'(x)$ for $f(x) =\displaystyle \cos ^4x $ .

    $f'(x) =\displaystyle 4\,\cos ^3x\,\sin x $ $f'(x) =\displaystyle 4\,\cos x\,\sin ^3x $ $f'(x) =\displaystyle -4\,\cos x\,\sin ^3x $ $f'(x) =\displaystyle -4\,\cos ^3x\,\sin x $

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

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

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

    $f'(x) =\displaystyle 2\,x\,\left(\sqrt{x^2+1}+1\right)^3 $ $f'(x) =\displaystyle {{3\,x\,\left(\sqrt{x^2+1}+1\right)^2}\over{\sqrt{x^2+1}}} $ $f'(x) =\displaystyle {{x\,\left(\sqrt{x^2+1}+1\right)^3}\over{\sqrt{x^2+1}}} $ $f'(x) =\displaystyle 6\,x\,\left(\sqrt{x^2+1}+1\right)^3 $ $f'(x) =\displaystyle {{3\,\sqrt{x^2+1}\,\left(\sqrt{x^2+1}+1\right)^2}\over{2}} $ $f'(x) =\displaystyle {{3\,\left(\sqrt{x^2+1}+1\right)^2}\over{2\,\sqrt{x^2+1}}} $

  10. Find the derivative $f'(x)$ for $f(x) =\displaystyle \left(\cos ^3x+2\right)^5 $ .

    $f'(x) =\displaystyle -\left(\cos ^3x+2\right)^5\,\sin x $ $f'(x) =\displaystyle -5\,\left(\cos ^3x+2\right)^5\,\sin x $ $f'(x) =\displaystyle -3\,\cos ^2x\,\left(\cos ^3x+2\right)^5\,\sin x $ $f'(x) =\displaystyle 15\,\cos ^3x\,\left(\cos ^3x+2\right)^4 $ $f'(x) =\displaystyle 15\,\cos ^2x\,\left(\cos ^3x+2\right)^4 $ $f'(x) =\displaystyle -15\,\cos ^2x\,\left(\cos ^3x+2\right)^4\,\sin x $


Department of Mathematics
Last modified: 2026-03-24