対称微分
絶対値関数 f ( x ) = | x | {\displaystyle f(x)=\left\vert x\right\vert } は x = 0 {\displaystyle x=0} において、
f s ( 0 ) = lim h → 0 f ( 0 + h ) − f ( 0 − h ) 2 h = lim h → 0 f ( h ) − f ( − h ) 2 h = lim h → 0 | h | − | − h | 2 h = lim h → 0 | h | − | h | 2 h = 0 {\displaystyle {\begin{aligned}f_{s}(0)&=\lim _{h\to 0}{\frac {f(0+h)-f(0-h)}{2h}}\\&=\lim _{h\to 0}{\frac {f(h)-f(-h)}{2h}}\\&=\lim _{h\to 0}{\frac {\left\vert h\right\vert -\left\vert -h\right\vert }{2h}}\\&=\lim _{h\to 0}{\frac {\left\vert h\right\vert -\left\vert h\right\vert }{2h}}\\&=0\end{aligned}}}