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

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

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

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

  3. Find the derivative $f'(x)$ for $f(x) =\displaystyle {{1}\over{5^{x}}} $ .

    $f'(x) =\displaystyle {{1}\over{5^{x}}} $ $f'(x) =\displaystyle -{{\ln 5}\over{5^{x}}} $ $f'(x) =\displaystyle x\,5^{1-x} $ $f'(x) =\displaystyle -\ln 5\,e^{x} $ $f'(x) =\displaystyle {{\ln 5}\over{5^{x}}} $

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

    $f'(x) =\displaystyle 3\,x^2\,e^{x}\,\sec x^3\,\tan x^3+e^{x}\,\sec x^3 $ $f'(x) =\displaystyle -3\,x^2\,e^{x}\,\sec x^3\,\tan x^3-e^{x}\,\sec x^3 $ $f'(x) =\displaystyle e^{x}\,\csc x^3-3\,x^2\,e^{x}\,\cot x^3\,\csc x^3 $ $f'(x) =\displaystyle 3\,x^2\,e^{x}\,\cot x^3\,\csc x^3-e^{x}\,\csc x^3 $

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

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

  6. Find the derivative $f'(x)$ for $f(x) =\displaystyle \left( {{x-1}\over{x+1}} \right)^{ {{2}\over{3}}}$ .

    $f'(x) =\displaystyle {{2\,x^{{{2}\over{3}}}}\over{3\,\left(x-1\right)^{{{1}\ove... ...{2}\over{3}}}\,x^{{{2}\over{3}}}}\over{3\, \left(x+1\right)^{{{5}\over{3}}}}} $ $f'(x) =\displaystyle {{\left(x-1\right)^{{{5}\over{3}}}}\over{\left(x+1\right)^{{{5 }\over{3}}}}} $ $f'(x) =\displaystyle {{2\,\left(x-1\right)^{{{2}\over{3}}}}\over{\left(x+1\right)^{{{8 }\over{3}}}}} $ $f'(x) =\displaystyle {{4}\over{3\,\left(x-1\right)^{{{1}\over{3}}}\,\left(x+1\right)^{{{ 5}\over{3}}}}} $ $f'(x) =\displaystyle {{2\,\left(x+1\right)^{{{1}\over{3}}}}\over{3\,\left(x-1\right)^{{{ 1}\over{3}}}}} $

  7. Find the derivative $f'(x)$ for $f(x) =\displaystyle \left(\left({{1}\over{\sqrt{x}}}+1\right)^2+1\right)^2 $ .

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

  8. Find the derivative $f'(x)$ for $f(x) =\displaystyle {{1}\over{2^{\sin x}}} $ .

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

  9. Find the derivative $f'(x)$ for $f(x) =\displaystyle \left(\sin ^4x+2\right)^3 $ .

    $f'(x) =\displaystyle 4\,\cos x\,\sin ^3x\,\left(\sin ^4x+2\right)^3 $ $f'(x) =\displaystyle \cos x\,\left(\sin ^4x+2\right)^3 $ $f'(x) =\displaystyle 3\,\cos x\,\left(\sin ^4x+2\right)^3 $ $f'(x) =\displaystyle 12\,\sin ^3x\,\left(\sin ^4x+2\right)^2 $ $f'(x) =\displaystyle 12\,\cos x\,\sin ^3x\,\left(\sin ^4x+2\right)^2 $ $f'(x) =\displaystyle 12\,\sin ^4x\,\left(\sin ^4x+2\right)^2 $

  10. Find the derivative $f'(x)$ for $f(x) =\displaystyle \sqrt{x+\sqrt{x}} $ .

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


Department of Mathematics
Last modified: 2026-05-18