Answer
$\dfrac {4\arcsin 2x}{\sqrt {1-4x^{2}}}$
Work Step by Step
$\dfrac {d}{dx}\left( \arcsin 2x\right) ^{2}=2\arcsin 2x\times \left( \dfrac {d}{dx}\left( \arcsin 2x\right) \right) =\dfrac {2\arcsin 2x}{\sqrt {1-\left( 2x\right) ^{2}}}\times \dfrac {d}{dx}\left( 2x\right) =\dfrac {4\arcsin 2x}{\sqrt {1-4x^{2}}}$
