(1)

\begin{align*} \frac{d\sgn\left(x\right)}{dx} & =\frac{d}{dx}\left(2H_{\frac{1}{2}}\left(x)-1\right)\right)\\ & =2\delta\left(x\right) \end{align*}

(2)

\begin{align*} \int\sgn\left(x\right)dx & =\int_{0}^{x}\left(2H_{\frac{1}{2}}\left(x)-1\right)\right)dx+C\\ & =\left[2\max\left(0,x\right)-x\right]_{0}^{x}+C\\ & =2\max\left(0,x\right)-x+C\\ & =\max\left(-x,x\right)+C\\ & =\left|x\right|+C \end{align*}

(2)-2

\begin{align*} \int\sgn\left(x\right)dx & =\int_{0}^{x}\sgn\left(x\right)dx+C\\ & =C+\begin{cases} x & 0\leq x\\ -x & x<0 \end{cases}\\ & =\left|x\right|+C \end{align*}