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

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

  2. Find the derivative $f'(x)$ for $f(x) =\displaystyle 2^{\cos x} $ .

    $f'(x) =\displaystyle \ln 2\,2^{\cos x}\,\cos x $ $f'(x) =\displaystyle -\ln 2\,2^{\cos x}\,\sin x $ $f'(x) =\displaystyle \ln 2\,2^{\cos x} $ $f'(x) =\displaystyle 2^{\cos x}\,\cos ^2x $ $f'(x) =\displaystyle -2^{\cos x}\,\cos x\,\sin x $

  3. Find the derivative $f'(x)$ for $f(x) =\displaystyle \cos ^3x $ .

    $f'(x) =\displaystyle 3\,\cos ^2x\,\sin x $ $f'(x) =\displaystyle -3\,\cos ^2x\,\sin x $ $f'(x) =\displaystyle 3\,\cos x\,\sin ^2x $ $f'(x) =\displaystyle -3\,\cos x\,\sin ^2x $

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

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

  5. Find the derivative $f'(x)$ for $f(x) =\displaystyle \cot \cos x $ .

    $f'(x) =\displaystyle \cot x\,\sin x\,\left(\csc \cos x\right)^2-\left(\csc x\right)^2\, \cot \cos x $ $f'(x) =\displaystyle \sin x\,\left(\csc \cos x\right)^2 $ $f'(x) =\displaystyle -\sin x\,\cot \cos x $ $f'(x) =\displaystyle -\left(\csc \cos x\right)^2 $ $f'(x) =\displaystyle \cos x\,\cot \cos x $

  6. Suppose that $g( 2 ) = 5 $ and $g'( 2 ) = 6 $. Then find the derivative $f'( 2 )$ for $f(x) = \displaystyle \sqrt{g\left(x\right)+1} $ .

    $\displaystyle {{3}\over{\sqrt{6}}}$ $\displaystyle {{5}\over{2\,\sqrt{6}}}$ $\displaystyle {{1}\over{2\,\sqrt{6}}}$ $\displaystyle \sqrt{6}$

  7. Find the derivative $f'(x)$ for $f(x) =\displaystyle \left(\cos ^3x+2\right)^3 $ .

    $f'(x) =\displaystyle -\left(\cos ^3x+2\right)^3\,\sin x $ $f'(x) =\displaystyle 9\,\cos ^3x\,\left(\cos ^3x+2\right)^2 $ $f'(x) =\displaystyle -3\,\left(\cos ^3x+2\right)^3\,\sin x $ $f'(x) =\displaystyle -9\,\cos ^2x\,\left(\cos ^3x+2\right)^2\,\sin x $ $f'(x) =\displaystyle -3\,\cos ^2x\,\left(\cos ^3x+2\right)^3\,\sin x $ $f'(x) =\displaystyle 9\,\cos ^2x\,\left(\cos ^3x+2\right)^2 $

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

    $\displaystyle -3\,e^ {- 6 }$ $\displaystyle e^ {- 6 }$ $\displaystyle -6\,e^ {- 6 }$ $\displaystyle 6\,e^ {- 6 }$

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

    $f'(x) =\displaystyle 3\,x^2\,e^{x^3} $ $f'(x) =\displaystyle x^3\,e^{x}+3\,x^2\,e^{x} $ $f'(x) =\displaystyle x^3\,e^{x} $ $f'(x) =\displaystyle 3\,x^2\,e^{x} $ $f'(x) =\displaystyle x^3\,e^{x^3} $

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

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


Department of Mathematics
Last modified: 2026-08-30