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

  3. Find the equation of the tangent line to $y = 3\,x^2-3\,x^{{{3}\over{2}}}+2 $ at $x = 4 $ .

    y = 15x − 42 y = 17x − 34 y = 15x − 34 y = 17x − 42

  4. Suppose that $g( 9 ) = -2 $ and $g'( 9 ) = -2 $. Then find the derivative $f'( 9 )$ for $f(x) = \displaystyle x^{{{3}\over{2}}}\,g\left(x\right) $ .

    −54 −63 $\displaystyle -2\,x^{{{3}\over{2}}}$ $\displaystyle -2\,x^{{{3}\over{2}}}-3\,\sqrt{x}$ −2 $\displaystyle -x^{{{3}\over{2}}}$

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

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

  6. Suppose that $g'( 8 ) = 2 $. Then find the derivative $f'( 8 )$ for $f(x) = -3\,g\left(x\right)-3\,x^{{{5}\over{3}}} $ .

    −20 −18 −26 −96

  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 $\displaystyle {{5\,a^{{{2}\over{3}}}}\over{3}}$ $\displaystyle a^{{{5}\over{3}}}$ $\displaystyle -{{5\,a^{{{2}\over{3}}}}\over{3}}$ $\displaystyle -{{5\,a^{{{5}\over{3}}}}\over{3}}$

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

    $f'(x) =\displaystyle -{{3}\over{\sqrt{x}}}-{{4}\over{x^3}} $ $f'(x) =\displaystyle -3\,\sqrt{x}-{{4}\over{x^2}} $ $f'(x) =\displaystyle -{{3}\over{2\,\sqrt{x}}}+{{8}\over{x^3}}+1 $ $f'(x) =\displaystyle {{8}\over{x^3}}-{{3}\over{2\,\sqrt{x}}} $

  9. Suppose that $g( 3 ) = 1 $ and $g'( 3 ) = -2 $. Then find the equation of the tangent line to $y = \displaystyle {{g\left(x\right)}\over{x-1}} $ at $x = 3 $ .

    $\displaystyle y={{11}\over{4}}-{{5\,x}\over{4}}$ $\displaystyle y={{17}\over{4}}-{{3\,x}\over{4}}$ $\displaystyle y={{17}\over{4}}-{{5\,x}\over{4}}$ $\displaystyle y={{11}\over{4}}-{{3\,x}\over{4}}$

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

    $\displaystyle {{x^2+2}\over{4}}$ $\displaystyle -{{23}\over{4}}$ $\displaystyle -{{11}\over{2}}$ $\displaystyle -{{x^2+2}\over{2}}$ $\displaystyle -{{x^2+2}\over{4}}-x$ $\displaystyle {{23}\over{2}}$


Department of Mathematics
Last modified: 2025-08-29