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

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

  3. Suppose that $g( 9 ) = 1 $ and $g'( 9 ) = -1 $. Then find the derivative $f'( 9 )$ for $f(x) = \displaystyle \sqrt{x}\,g\left(x\right) $ .

    −1 $\displaystyle -{{17}\over{6}}$ $\displaystyle \sqrt{x}$ 3 $\displaystyle {{1}\over{2\,\sqrt{x}}}-\sqrt{x}$ $\displaystyle 2\,\sqrt{x}$

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

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

  5. Find the derivative $f'(x)$ for $f(x) = \displaystyle {{1}\over{x^2}} + \frac{1} { \sqrt[ 4 ]{x^{ 3 } } }$ .

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

  6. Find the equation of the tangent line to $y = -1 $ at $x = 4 $ .

    y = −x − 1 y = 3 y = −1 y = 3 − x

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

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

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

    y = −3x − 4 y = −x − 4 y = −x − 2 y = −3x − 2

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

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

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

    $\displaystyle {{x^2+3}\over{2}}$ $\displaystyle {{x^2+3}\over{4}}$ 11 $\displaystyle {{11}\over{2}}$ $\displaystyle {{x^2+3}\over{2}}+x$ $\displaystyle {{7}\over{2}}$


Department of Mathematics
Last modified: 2026-10-24