Generating...                               s20quiz08_n21

  1. Find the derivative $f'(x)$ for $f(x) = 3\,x-3\,\sqrt{x}+1 $ .

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

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

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

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

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

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

    $f'(x) =\displaystyle -16\,x^{{{7}\over{2}}}-6\,\sqrt{x} $ $f'(x) =\displaystyle -8\,x^{{{5}\over{2}}}-{{6}\over{\sqrt{x}}} $ $f'(x) =\displaystyle -8\,x^{{{5}\over{2}}}-{{3}\over{\sqrt{x}}} $ $f'(x) =\displaystyle -8\,x^2-{{3}\over{x}} $ $f'(x) =\displaystyle -7\,x^{{{7}\over{2}}}-{{3\,\sqrt{x}}\over{2}}+{{3}\over{2\,\sqrt{x} }} $ $f'(x) =\displaystyle -7\,x^{{{5}\over{2}}}-{{3}\over{2\,\sqrt{x}}}+{{3}\over{2\,x^{{{3 }\over{2}}}}} $

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

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

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

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

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

    −1 $\displaystyle -{{2}\over{3}}$ $\displaystyle {{4}\over{3}}$ $\displaystyle {{4}\over{3}}$

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

    $\displaystyle y={{7\,x}\over{4}}+5$ $\displaystyle y={{x}\over{4}}+2$ $\displaystyle y={{7\,x}\over{4}}+2$ $\displaystyle y={{x}\over{4}}+5$

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

    $\displaystyle y={{3}\over{2}}-{{21\,x}\over{2}}$ $\displaystyle y={{3}\over{2}}-{{17\,x}\over{2}}$ $\displaystyle y={{7}\over{2}}-{{21\,x}\over{2}}$ $\displaystyle y={{7}\over{2}}-{{17\,x}\over{2}}$


Department of Mathematics
Last modified: 2026-05-20