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\mbox{This basically will need 3 ingredients:} \ \ \mbox{product rule, arctan rule, simplification.}
\mbox{We are given:} \qquad y = x^2 arctan(x).
\mbox{Thus:}
\mbox{1) product rule:} \qquad {dy}/{dx} = x^2 d/{dx} [ arctan(x) ] + arctan(x) d/{dx} [ x^2 ]
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = x^2 d/{dx} [ arctan(x) ] + [ arctan(x) ][ 2x ]
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = x^2 d/{dx} [ arctan(x) ] + 2x arctan(x).
\mbox{2) arctan rule:} \qquad \quad {dy}/{dx} = x^2 ( 1 / { x^2 + 1 } ) + 2x arctan(x).
\mbox{3) simplification:} \quad {dy}/{dx} = x^2 / { x^2 + 1 } + { 2x ( x^2 + 1 ) arctan(x) } / { x^2 + 1 }
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = { x^2 + 2x ( x^2 + 1 ) arctan(x) } / { x^2 + 1 }
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = { x^2 + ( 2 x^3 + 2 x ) arctan(x) } / { x^2 + 1 }.
\mbox{4) Final Answer:} \quad {dy}/{dx} = { x^2 + ( 2 x^3 + 2 x ) arctan(x) } / { x^2 + 1 }.
