Generating...                               s20quiz08_n4

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

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

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

    $f'(x) =\displaystyle 10\,x^{{{5}\over{2}}}+3\,x^{{{3}\over{2}}}+{{3}\over{2\,\sqrt{x}}} $ $f'(x) =\displaystyle 24\,x^{{{5}\over{2}}}+8\,x^{{{3}\over{2}}} $ $f'(x) =\displaystyle 10\,x^{{{3}\over{2}}}+3\,\sqrt{x}+{{3}\over{2\,x^{{{3}\over{2}}}}} $ $f'(x) =\displaystyle 12\,x^{{{3}\over{2}}}+4\,\sqrt{x}-{{3}\over{\sqrt{x}}} $ $f'(x) =\displaystyle 12\,x^{{{3}\over{2}}}+4\,\sqrt{x} $ $f'(x) =\displaystyle 12\,x+4 $

  3. Find the derivative $f'(x)$ for $f(x) = 2\,x^{{{5}\over{4}}}-{{1}\over{x^2}}+1 $ .

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

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

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

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

    −2 0 −6 −3

  6. Find the equation of the tangent line to $y = -2\,x^2-3\,x^{{{3}\over{2}}}+1 $ at $x = 1 $ .

    $\displaystyle y={{7}\over{2}}-{{15\,x}\over{2}}$ $\displaystyle y={{7}\over{2}}-{{17\,x}\over{2}}$ $\displaystyle y={{9}\over{2}}-{{15\,x}\over{2}}$ $\displaystyle y={{9}\over{2}}-{{17\,x}\over{2}}$

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

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

  8. Suppose that $g( 9 ) = -2 $ and $g'( 9 ) = -2 $. Then find the derivative $f'( 9 )$ for $f(x) = \displaystyle \sqrt{x}\,g\left(x\right) $ .

    $\displaystyle -{{19}\over{3}}$ $\displaystyle -2\,\sqrt{x}-{{1}\over{\sqrt{x}}}$ $\displaystyle -\sqrt{x}$ −6 $\displaystyle -2\,\sqrt{x}$ −2

  9. Find the derivative $f'(x)$ for $f(x) = \displaystyle {{4-2\,x}\over{x-3}} $ .

    $f'(x) = \displaystyle -{{2}\over{x^2-6\,x+9}} $ $f'(x) = \displaystyle {{2}\over{x^2-6\,x+9}} $ $f'(x) = \displaystyle {{3\,x}\over{x^2-6\,x+9}} $ $f'(x) = \displaystyle -{{3\,x}\over{x^2-6\,x+9}} $

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

    y = 2x + 3 y = 2x + 5 y = x + 3 y = x + 5


Department of Mathematics
Last modified: 2025-06-19