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

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

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

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

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

    $\displaystyle -{{3}\over{25}}$ $\displaystyle {{1}\over{5}}$ $\displaystyle -{{2}\over{25}}$ $\displaystyle -{{1}\over{25}}$

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

    $f'(x) =\displaystyle -{{\sin x}\over{2\,\left(\cos x\right)^{{{3}\over{2}}}}} $ $f'(x) =\displaystyle {{\cos x}\over{2\,\left(\sin x\right)^{{{3}\over{2}}}}} $ $f'(x) =\displaystyle -{{\cos x}\over{2\,\left(\sin x\right)^{{{3}\over{2}}}}} $ $f'(x) =\displaystyle {{\sin x}\over{2\,\left(\cos x\right)^{{{3}\over{2}}}}} $

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

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

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

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

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

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

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

  9. Find the derivative $f'(x)$ for $f(x) = \displaystyle \sqrt[ 4 ]{ \left(x^4+2\,x+3\right)^3 }$ .

    $f'(x) =\displaystyle {{3\,\left(4\,x^3+2\right)}\over{4\,\left(x^4+2\,x+3\right)^{{{1 }\over{4}}}}} $ $f'(x) =\displaystyle \left(x^4+2\,x+3\right)^{{{7}\over{4}}} $ $f'(x) =\displaystyle \left(4\,x^3+2\right)\,\left(x^4+2\,x+3\right)^{{{3}\over{4}}} $ $f'(x) =\displaystyle {{3}\over{4\,\left(x^4+2\,x+3\right)^{{{1}\over{4}}}}} $ $f'(x) =\displaystyle {{3\,\left(x^4+2\,x+3\right)^{{{3}\over{4}}}}\over{4\,x^{{... ...}}\,\left(4\,x^3+2\right)}\over{4\,\left( x^4+2\,x+3\right)^{{{1}\over{4}}}}} $

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

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


Department of Mathematics
Last modified: 2025-11-27