Generating...                               quiz12_n21

  1. Find the derivative $f'(x)$ for $f(x) =\displaystyle e^x \tan x^2 $ .

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

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

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

  3. Find the derivative $f'(x)$ for $f(x) =\displaystyle 5^{\cos x} $ .

    $f'(x) =\displaystyle \ln 5\,5^{\cos x} $ $f'(x) =\displaystyle \ln 5\,5^{\cos x}\,\cos x $ $f'(x) =\displaystyle 5^{\cos x}\,\cos ^2x $ $f'(x) =\displaystyle -\ln 5\,5^{\cos x}\,\sin x $ $f'(x) =\displaystyle -5^{\cos x}\,\cos x\,\sin x $

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

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

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

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

  6. Find the derivative $f'(x)$ for $f(x) =\displaystyle {{1}\over{2^{x}}} $ .

    $f'(x) =\displaystyle -{{\ln 2}\over{2^{x}}} $ $f'(x) =\displaystyle {{1}\over{2^{x}}} $ $f'(x) =\displaystyle x\,2^{1-x} $ $f'(x) =\displaystyle {{\ln 2}\over{2^{x}}} $ $f'(x) =\displaystyle -\ln 2\,e^{x} $

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

    $f'(x) =\displaystyle 3\,x^4\,e^{x^3} $ $f'(x) =\displaystyle x^2\,e^{x}+2\,x\,e^{x} $ $f'(x) =\displaystyle 6\,x^3\,e^{x^3} $ $f'(x) =\displaystyle 3\,x^4\,e^{x^3}+2\,x\,e^{x^3} $ $f'(x) =\displaystyle x^3\,\left(x^2\,e^{x}+2\,x\,e^{x}\right) $

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

    $f'(x) =\displaystyle \left(x^3+x+1\right)^{{{7}\over{4}}} $ $f'(x) =\displaystyle {{3\,\left(x^3+x+1\right)^{{{3}\over{4}}}}\over{4\,x^{{{1}... ...{4}}}\,\left(3\,x^2+1\right)}\over{4\,\left(x^3 +x+1\right)^{{{1}\over{4}}}}} $ $f'(x) =\displaystyle {{3}\over{4\,\left(x^3+x+1\right)^{{{1}\over{4}}}}} $ $f'(x) =\displaystyle \left(3\,x^2+1\right)\,\left(x^3+x+1\right)^{{{3}\over{4}}} $ $f'(x) =\displaystyle {{3\,\left(3\,x^2+1\right)}\over{4\,\left(x^3+x+1\right)^{{{1 }\over{4}}}}} $

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

    $\displaystyle -{{2}\over{125}}$ $\displaystyle -{{2}\over{25}}$ $\displaystyle {{1}\over{25}}$ $\displaystyle -{{6}\over{125}}$

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

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


Department of Mathematics
Last modified: 2025-09-14