$($i$)$ Differentiate and simplify:

$f(x)=e^{xsin2x}.$

$($ii$)$ Solve for $x$ :

$ln(2x+1)=2-lnx.$

$($iii$)$ Find an equation of the tangent line to the curve $y=x^{2}lnx$ at $(1,0).$

SHOW ALL WORK