I know I can integrate $|x|$ using the the sign function $\operatorname{sgn}(x)$ as $\int|x|dx=\frac{x^2}{2}\operatorname{sgn}(x)+C$ where $\operatorname{sgn}(x)=\frac{x}{|x|}=\frac{d}{dx}|x|$. But when I differentiate $\frac{x^2}{2}\operatorname{sgn}(x)=\frac{x^3}{2|x|}$ I get
\frac{d}{dx}\frac{x^3}{2|x|}=\frac{3x^2(2|x|)+x^3(2\operatorname{sgn}(x))}{(2|x|)^2}=\frac{6x^2|x|+\frac{2x^4}{|x|}}{4x^2}=\frac{8x^4}{4x^2|x|}=2x\operatorname{sgn}(x)=\frac{2x^2}{|x|}=(2|x|)\frac{|x|}{|x|}$
Which implies 2|x|= |x|$, but we all know $1\ne2$. Where did I go wrong?