Symmetric derivative ($f^s(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x-h)}{2h}$)

The symmetric derivative compares values equally far to either side of a point. If the ordinary derivative exists, splitting the symmetric difference quotient into two ordinary difference quotients shows $f^s(x)=f^{\prime }(x)$. The converse fails: the absolute value function has $f^s(0)=0$ but distinct left and right one-sided derivatives at zero.