$f(x) = |x|$
$|x| = \begin{cases} x, & x \ge 0 \ -x, & x < 0 \end{cases}$
$f'(x) = \begin{cases} 1, & x > 0 \ -1, & x < 0 \end{cases}$
Left derivative at $x = 0 = -1$
Right derivative at $x = 0 = 1$
Derivative does not exist.
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and More.