I found some derivatives and integrals involving the absolute value functions are expressable in terms of the absolute value functions, for examples:
$\dfrac{d}{dx}|x|$
$=\lim\limits_{\Delta x\to 0}\dfrac{|x+\Delta x|-|x|}{\Delta x}$
$=\lim\limits_{\Delta x\to 0}\dfrac{(|x+\Delta x|-|x|)(|x+\Delta x|+|x|)}{\Delta x(|x+\Delta x|+|x|)}$
$=\lim\limits_{\Delta x\to 0}\dfrac{|x+\Delta x|^2-|x|^2}{\Delta x(|x+\Delta x|+|x|)}$
$=\lim\limits_{\Delta x\to 0}\dfrac{(x+\Delta x)^2-x^2}{\Delta x(|x+\Delta x|+|x|)}$
$=\lim\limits_{\Delta x\to 0}\dfrac{x^2+2x\Delta x+(\Delta x)^2-x^2}{\Delta x(|x+\Delta x|+|x|)}$
$=\lim\limits_{\Delta x\to 0}\dfrac{2x+\Delta x}{|x+\Delta x|+|x|}$
$=\dfrac{x}{|x|}$
$=\dfrac{x|x|}{|x|^2}$
$=\dfrac{x|x|}{x^2}$
$=\dfrac{|x|}{x}$
$\int|x|~dx$
$=x|x|-\int x~d(|x|)$
$=x|x|-\int x\times\dfrac{|x|}{x}dx$
$=x|x|-\int|x|~dx$
$\therefore\int|x|~dx=\dfrac{x|x|}{2}+C$
Why does current mathematics education often not to teach like these?