Generating...                               s20quiz08_n1

  1. Suppose that $g'( 27 ) = -3 $. Then find the derivative $f'( 27 )$ for $f(x) = 3\,x^{{{1}\over{3}}}-3\,g\left(x\right) $ .

    $\displaystyle -{{26}\over{9}}$ $\displaystyle {{82}\over{9}}$ 9 $\displaystyle {{1}\over{9}}$

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

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

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

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

  4. 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 $ .

    $\displaystyle y={{9\,x}\over{4}}+6$ $\displaystyle y={{3\,x}\over{4}}+3$ $\displaystyle y={{9\,x}\over{4}}+3$ $\displaystyle y={{3\,x}\over{4}}+6$

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

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

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

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

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

    $\displaystyle x^2+3$ $\displaystyle -x^2-3$ −7 18 −18 $\displaystyle -2\,\left(x^2+3\right)-2\,x$

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

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

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

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

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

    $\displaystyle 2\,\sqrt{x}$ $\displaystyle \sqrt{x}$ −2 $\displaystyle {{1}\over{2\,\sqrt{x}}}-2\,\sqrt{x}$ $\displaystyle -{{15}\over{4}}$ 2


Department of Mathematics
Last modified: 2026-07-16