1. Suppose that $g( -1 ) = 2 $ and $g'( -1 ) = 3 $. Then find the derivative $f'( -1 )$ for $f(x) = \displaystyle e^{3\,g\left(x\right)} $ .

    $\displaystyle 3\,e^6$ $\displaystyle 6\,e^6$ $\displaystyle 9\,e^6$ $\displaystyle e^6$

  2. 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 -{{\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}}} $

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

    $f'(x) =\displaystyle \ln 2\,2^{\tan x}\,\left(\sec x\right)^2 $ $f'(x) =\displaystyle \ln 2\,2^{\tan x} $ $f'(x) =\displaystyle 2^{\tan x}\,\tan ^2x $ $f'(x) =\displaystyle 2^{\tan x}\,\left(\sec x\right)^2\,\tan x $ $f'(x) =\displaystyle \ln 2\,2^{\tan x}\,\tan x $

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

    $f'(x) =\displaystyle 6\,x^2\,\left(x^3+1\right)\,\left(\left(x^3+1\right)^2+1\right)^2 $ $f'(x) =\displaystyle 12\,x^2\,\left(x^3+1\right)\,\left(\left(x^3+1\right)^2+1\right) $ $f'(x) =\displaystyle 3\,x^2\,\left(\left(x^3+1\right)^2+1\right)^2 $ $f'(x) =\displaystyle 4\,\left(x^3+1\right)\,\left(\left(x^3+1\right)^2+1\right) $ $f'(x) =\displaystyle 4\,\left(x^3+1\right)^2\,\left(\left(x^3+1\right)^2+1\right) $ $f'(x) =\displaystyle 6\,x^2\,\left(\left(x^3+1\right)^2+1\right)^2 $

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

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

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

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

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

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

    $f'(x) =\displaystyle 3\,\cos x\,\sin ^2x\,\left(\sin ^3x+1\right)^4 $ $f'(x) =\displaystyle 12\,\cos x\,\sin ^2x\,\left(\sin ^3x+1\right)^3 $ $f'(x) =\displaystyle 12\,\sin ^2x\,\left(\sin ^3x+1\right)^3 $ $f'(x) =\displaystyle 12\,\sin ^3x\,\left(\sin ^3x+1\right)^3 $ $f'(x) =\displaystyle 4\,\cos x\,\left(\sin ^3x+1\right)^4 $ $f'(x) =\displaystyle \cos x\,\left(\sin ^3x+1\right)^4 $

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

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

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

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


Department of Mathematics
Last modified: 2026-05-25