Generating...                               s20quiz08_n19

  1. Suppose that $g( -1 ) = -1 $ and $g'( -1 ) = 3 $. Then find the equation of the tangent line to $y = \displaystyle {{g\left(x\right)}\over{x+3}} $ at $x = -1 $ .

    $\displaystyle y={{7\,x}\over{4}}+{{3}\over{4}}$ $\displaystyle y={{7\,x}\over{4}}+{{5}\over{4}}$ $\displaystyle y={{5\,x}\over{4}}+{{5}\over{4}}$ $\displaystyle y={{5\,x}\over{4}}+{{3}\over{4}}$

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

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

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

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

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

    $\displaystyle {{3}\over{4}}$ $\displaystyle -{{3}\over{2}}$ $\displaystyle -{{5}\over{2}}$ $\displaystyle {{x+2}\over{4}}-{{1}\over{2}}$ $\displaystyle {{x+2}\over{4}}$ $\displaystyle -{{x+2}\over{2}}$

  5. 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) $ .

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

  6. Find the derivative $f'(x)$ for $f(x) = \displaystyle {{e^{x}+4\,x}\over{e^{x}+3}} $.

    $f'(x) = \displaystyle -{{\left(4\,x-7\right)\,e^{x}-12}\over{e^{2\,x}+6\,e^{x}+9}} $ $f'(x) = \displaystyle -{{\left(4\,x-1\right)\,e^{x}-12}\over{e^{2\,x}+6\,e^{x}+9}} $ $f'(x) = \displaystyle {{\left(4\,x-1\right)\,e^{x}-12}\over{e^{2\,x}+6\,e^{x}+9}} $ $f'(x) = \displaystyle {{\left(4\,x-7\right)\,e^{x}-12}\over{e^{2\,x}+6\,e^{x}+9}} $

  7. Let $a > 0$. Then find $\displaystyle \lim_{h\rightarrow 0}{{{\left(h+a\right)^{{{5}\over{3}}}-a^{{{5 }\over{3}}}}\over{h}}}$ .

    $\displaystyle -{{5\,a^{{{5}\over{3}}}}\over{3}}$ Does not exist $\displaystyle {{5\,a^{{{2}\over{3}}}}\over{3}}$ $\displaystyle -{{5\,a^{{{2}\over{3}}}}\over{3}}$ $\displaystyle a^{{{5}\over{3}}}$ $\displaystyle {{5\,a^{{{5}\over{3}}}}\over{3}}$

  8. Suppose that $g'( 1 ) = -1 $. Then find the derivative $f'( 1 )$ for $f(x) = -3\,g\left(x\right)-4\,x^{{{1}\over{3}}} $ .

    $\displaystyle -{{7}\over{3}}$ $\displaystyle {{5}\over{3}}$ $\displaystyle -{{4}\over{3}}$ −4

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

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

  10. Find the derivative $f'(x)$ for $f(x) = \displaystyle {{2-3\,x}\over{x-1}} $ .

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


Department of Mathematics
Last modified: 2025-09-14