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Let $g(x)=e^{x^2} \cdot \tan^{-1}(2x)$. What is the slope of the tangent line at $x=\cfrac{1}{2}$?

A

$\cfrac{\sqrt[4]{e} (4+\pi)}{4}$

B

$\sqrt[4]{e}$

C

$\cfrac{\sqrt[4]{e}(2+\pi)}{4}$

D

$\cfrac{\sqrt[4]{e}(16+3\pi)}{12}$

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