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

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

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

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

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

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

  4. Suppose that $g( 5 ) = 6 $ and $g'( 5 ) = 3 $. Then find the derivative $f'( 5 )$ for $f(x) = \displaystyle {{1}\over{\left(g\left(x\right)+3\right)^2}} $ .

    $\displaystyle -{{2}\over{243}}$ $\displaystyle -{{2}\over{729}}$ $\displaystyle -{{4}\over{243}}$ $\displaystyle {{1}\over{81}}$

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

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

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

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

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

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

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

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

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

    $f'(x) =\displaystyle 4\,x^3\,e^{x}\,\cot x^4\,\csc x^4-e^{x}\,\csc x^4 $ $f'(x) =\displaystyle 4\,x^3\,e^{x}\,\sec x^4\,\tan x^4+e^{x}\,\sec x^4 $ $f'(x) =\displaystyle -4\,x^3\,e^{x}\,\sec x^4\,\tan x^4-e^{x}\,\sec x^4 $ $f'(x) =\displaystyle e^{x}\,\csc x^4-4\,x^3\,e^{x}\,\cot x^4\,\csc x^4 $

  10. Find the derivative $f'(x)$ for $f(x) = \displaystyle \sqrt[ 3 ]{ x^2+2\,x+3 }$ .

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


Department of Mathematics
Last modified: 2023-08-24