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) $ .

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

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

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

  3. Find the derivative $f'(x)$ for $f(x) = \displaystyle {{4\,x+10}\over{x+3}} $ .

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

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

    $\displaystyle y={{9}\over{4}}-{{5\,x}\over{4}}$ $\displaystyle y={{x}\over{4}}-{{9}\over{4}}$ $\displaystyle y={{x}\over{4}}+{{9}\over{4}}$ $\displaystyle y=-{{5\,x}\over{4}}-{{9}\over{4}}$

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

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

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

    $f'(x) =\displaystyle e^{x}+3 $ $f'(x) =\displaystyle 3\,x\,e^{x}+3\,e^{x} $ $f'(x) =\displaystyle x\,e^{x}+e^{x} $ $f'(x) =\displaystyle 3\,e^{x} $

  7. Find the derivative $f'(x)$ for $f(x) = \displaystyle {{1}\over{x^3}} + \frac{1} { \sqrt[ 4 ]{x^{ 7 } } }$ .

    $f'(x) =\displaystyle -{{3}\over{4\,x^{{{7}\over{4}}}}}-{{3}\over{x^4}} $ $f'(x) =\displaystyle {{7\,x^{{{3}\over{4}}}}\over{4}}-{{3}\over{x^4}} $ $f'(x) =\displaystyle {{3}\over{4\,x^{{{1}\over{4}}}}}-{{3}\over{x^4}} $ $f'(x) =\displaystyle -{{7}\over{4\,x^{{{11}\over{4}}}}}-{{3}\over{x^4}} $

  8. 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) $ .

    $\displaystyle {{17}\over{4}}$ 2 $\displaystyle 2\,\sqrt{x}$ $\displaystyle 2\,\sqrt{x}+{{1}\over{2\,\sqrt{x}}}$ 2 $\displaystyle \sqrt{x}$

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

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

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

    $f'(x) =\displaystyle {{1}\over{x}}-12\,x^2 $ $f'(x) =\displaystyle -12\,x-{{2}\over{x^2}} $ $f'(x) =\displaystyle -9\,x^2-{{3}\over{x^2}} $ $f'(x) =\displaystyle -12\,x^2-{{2}\over{x}} $ $f'(x) =\displaystyle -9\,x^3-{{3}\over{x}} $ $f'(x) =\displaystyle -12\,x^3-2 $


Department of Mathematics
Last modified: 2025-10-13