1. Find the derivative $f'(x)$ for $f(x) =\displaystyle \sin \tan x $ .

    $f'(x) =\displaystyle \left(\sec x\right)^2\,\sin \tan x $ $f'(x) =\displaystyle \tan x\,\sin \tan x $ $f'(x) =\displaystyle \cos \tan x $ $f'(x) =\displaystyle \left(\sec x\right)^2\,\cos \tan x $ $f'(x) =\displaystyle \cos x\,\sin \tan x+\left(\sec x\right)^2\,\sin x\,\cos \tan x $

  2. Find the derivative $f'(x)$ for $f(x) =\displaystyle x^2\,e^{x^3} $ .

    $f'(x) =\displaystyle 3\,x^4\,e^{x^3}+2\,x\,e^{x^3} $ $f'(x) =\displaystyle x^2\,e^{x}+2\,x\,e^{x} $ $f'(x) =\displaystyle x^3\,\left(x^2\,e^{x}+2\,x\,e^{x}\right) $ $f'(x) =\displaystyle 3\,x^4\,e^{x^3} $ $f'(x) =\displaystyle 6\,x^3\,e^{x^3} $

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

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

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

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

  5. Find the derivative $f'(x)$ for $f(x) = \displaystyle \sqrt[ 4 ]{ x^3+2\,x+1 }$ .

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

  6. Find the derivative $f'(x)$ for $f(x) =\displaystyle 5^{\sin x} $ .

    $f'(x) =\displaystyle 5^{\sin x}\,\cos x\,\sin x $ $f'(x) =\displaystyle \ln 5\,5^{\sin x}\,\sin x $ $f'(x) =\displaystyle 5^{\sin x}\,\sin ^2x $ $f'(x) =\displaystyle \ln 5\,5^{\sin x} $ $f'(x) =\displaystyle \ln 5\,5^{\sin x}\,\cos x $

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

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

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

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

  9. Find the derivative $f'(x)$ for $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}\over{3^{x}}} $ $f'(x) =\displaystyle {{1}\over{3^{x}}} $ $f'(x) =\displaystyle -\ln 3\,e^{x} $

  10. Find the derivative $f'(x)$ for $f(x) =\displaystyle \left(\cos ^3x+2\right)^5 $ .

    $f'(x) =\displaystyle -5\,\left(\cos ^3x+2\right)^5\,\sin x $ $f'(x) =\displaystyle -3\,\cos ^2x\,\left(\cos ^3x+2\right)^5\,\sin x $ $f'(x) =\displaystyle 15\,\cos ^3x\,\left(\cos ^3x+2\right)^4 $ $f'(x) =\displaystyle -15\,\cos ^2x\,\left(\cos ^3x+2\right)^4\,\sin x $ $f'(x) =\displaystyle -\left(\cos ^3x+2\right)^5\,\sin x $ $f'(x) =\displaystyle 15\,\cos ^2x\,\left(\cos ^3x+2\right)^4 $


Department of Mathematics
Last modified: 2025-09-22