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

  2. Suppose that $g'( 27 ) = 3 $. Then find the derivative $f'( 27 )$ for $f(x) = 4\,g\left(x\right)+x^{{{2}\over{3}}} $ .

    $\displaystyle {{29}\over{9}}$ $\displaystyle {{2}\over{9}}$ $\displaystyle {{110}\over{9}}$ 9

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

    $f'(x) =\displaystyle 6\,x^{{{3}\over{2}}}-2\,\sqrt{x}+{{1}\over{\sqrt{x}}} $ $f'(x) =\displaystyle 5\,x^{{{3}\over{2}}}-{{3\,\sqrt{x}}\over{2}}-{{1}\over{2\,x^{{{3 }\over{2}}}}} $ $f'(x) =\displaystyle 6\,x-2 $ $f'(x) =\displaystyle 6\,x^{{{3}\over{2}}}-2\,\sqrt{x} $ $f'(x) =\displaystyle 12\,x^{{{5}\over{2}}}-4\,x^{{{3}\over{2}}} $ $f'(x) =\displaystyle 5\,x^{{{5}\over{2}}}-{{3\,x^{{{3}\over{2}}}}\over{2}}-{{1}\over{2\, \sqrt{x}}} $

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

    $f'(x) =\displaystyle 4\,\sqrt{x}-{{4}\over{x^3}} $ $f'(x) =\displaystyle 6\,\sqrt{x}+{{8}\over{x^3}}+1 $ $f'(x) =\displaystyle 6\,\sqrt{x}+{{8}\over{x^3}} $ $f'(x) =\displaystyle 4\,x^{{{3}\over{2}}}-{{4}\over{x^2}} $

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

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

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

    $f'(x) = \displaystyle {{\left(3\,x-5\right)\,e^{x}-6}\over{e^{2\,x}+4\,e^{x}+4}} $ $f'(x) = \displaystyle {{\left(3\,x-1\right)\,e^{x}-6}\over{e^{2\,x}+4\,e^{x}+4}} $ $f'(x) = \displaystyle -{{\left(3\,x-5\right)\,e^{x}-6}\over{e^{2\,x}+4\,e^{x}+4}} $ $f'(x) = \displaystyle -{{\left(3\,x-1\right)\,e^{x}-6}\over{e^{2\,x}+4\,e^{x}+4}} $

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

    y = 5x − 6 y = 5x + 2 y = 3x − 6 y = 3x + 2

  8. Suppose that $g( -3 ) = -2 $ and $g'( -3 ) = -1 $. Then find the equation of the tangent line to $y = \displaystyle {{g\left(x\right)}\over{x+5}} $ at $x = -3 $ .

    y = −1 y = −4 y = −x − 4 y = −x − 1

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

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

  10. Let $a > 0$. Then find $\displaystyle \lim_{h\rightarrow 0}{{{\left(h+a\right)^{{{2}\over{3}}}-a^{{{2 }\over{3}}}}\over{h}}}$ .

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


Department of Mathematics
Last modified: 2026-09-08