Generating...                               s20quiz08_n14

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

    $f'(x) =\displaystyle {{e^{x}}\over{x^2}}-{{2\,e^{x}}\over{x^3}} $ $f'(x) =\displaystyle {{2\,e^{x}}\over{x^2}}-{{4\,e^{x}}\over{x^3}} $ $f'(x) =\displaystyle -{{4\,e^{x}}\over{x^3}} $ $f'(x) =\displaystyle e^{x}-{{4}\over{x^3}} $

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

    $f'(x) =\displaystyle 3\,x^2+{{3}\over{4\,x^{{{1}\over{4}}}}} $ $f'(x) =\displaystyle 3\,x^2+{{1}\over{4\,x^{{{3}\over{4}}}}} $ $f'(x) =\displaystyle 3\,x^2-{{3}\over{4\,x^{{{7}\over{4}}}}} $ $f'(x) =\displaystyle 3\,x^2-{{1}\over{4\,x^{{{5}\over{4}}}}} $

  3. Suppose that $g( -2 ) = 3 $ and $g'( -2 ) = -3 $. Then find the equation of the tangent line to $y = \displaystyle {{g\left(x\right)}\over{x}} $ at $x = -2 $ .

    $\displaystyle y={{9\,x}\over{4}}+3$ $\displaystyle y={{9\,x}\over{4}}$ $\displaystyle y={{3\,x}\over{4}}+3$ $\displaystyle y={{3\,x}\over{4}}$

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

    $f'(x) = -x\,e^{x-1}-{{\sqrt{x}}\over{2}} $ $f'(x) = -e^{x}-{{\sqrt{x}}\over{2}} $ $f'(x) = -e^{x}-{{x^{{{3}\over{2}}}}\over{2}} $ $f'(x) = -e^{x}-{{1}\over{2\,\sqrt{x}}} $

  5. Find the derivative $f'(x)$ for $f(x) = -2\,x^2+4\,x^{{{5}\over{4}}}-4 $ .

    $f'(x) =\displaystyle 5\,x^{{{1}\over{4}}}-4\,x $ $f'(x) =\displaystyle -4\,x+5\,x^{{{1}\over{4}}}-4 $ $f'(x) =\displaystyle 4\,x^{{{1}\over{4}}}-2\,x $ $f'(x) =\displaystyle 4\,x^{{{5}\over{4}}}-2\,x^2 $

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

    $\displaystyle -\sqrt{x}-{{1}\over{\sqrt{x}}}$ $\displaystyle -\sqrt{x}$ $\displaystyle -2\,\sqrt{x}$ −2 −2 −1

  7. Find the equation of the tangent line to $y = x^2+1 $ at $x = 1 $ .

    y = 2x y = 2x − 1 y = 3x − 1 y = 3x

  8. Let $a > 0$. Then find $\displaystyle \lim_{h\rightarrow 0}{{{\left(h+a\right)^{{{1}\over{3}}}-a^{{{1 }\over{3}}}}\over{h}}}$ .

    $\displaystyle a^{{{1}\over{3}}}$ $\displaystyle -{{1}\over{3\,a^{{{2}\over{3}}}}}$ $\displaystyle {{a^{{{1}\over{3}}}}\over{3}}$ Does not exist $\displaystyle {{1}\over{3\,a^{{{2}\over{3}}}}}$ $\displaystyle -{{a^{{{1}\over{3}}}}\over{3}}$

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

    −8 $\displaystyle x^2+2$ $\displaystyle -2\,\left(x^2+2\right)-2\,x$ −3 $\displaystyle -x^2-2$ 8

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

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


Department of Mathematics
Last modified: 2025-05-04