$\displaystyle
y = (1 + 4e^{2x})^{2x}
$
$\displaystyle
y' = (1 + 4e^{2x})^{2x} \left( \frac{16xe^{2x} + 8e^{2x}(1 + 4e^{2x})ln(1 + 4e^{2x})}{(1 + 4e^{2x})}\right)
$
$\displaystyle
y = \left( \frac{1}{x} \right)^{ln x}
$
$\displaystyle
y' = \left( \frac{1}{x} \right)^{ln x} \left(\frac{x^2lnx + ln \frac{1}{x}}{x}\right)
$
$\displaystyle
y = \frac {\sqrt[3]{x^3 + 1}}{x^3(x-1)^3}
$
$\displaystyle
y' = \frac {\sqrt[3]{x^3 + 1}}{x^3(x-1)^3} \left( \frac{x^2}{x^3+1} - \frac{3}{x} + \frac{3}{x-1} \right)
$
$\displaystyle
y = \frac {ln x}{e^{\sqrt {x}}cos^4x}
$
$\displaystyle
y' = \frac {ln x}{e^{\sqrt {x}}cos^4x} \left( \frac{1}{xlnx} - \frac{e^{\sqrt{x}}}{2 \sqrt{x} e^{\sqrt{x}}} - 4tanx \right)
$