Symmetric derivative

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In mathematics, the symmetric derivative is an operation related to the ordinary derivative.

It is defined as:

$\lim_{h \to 0}\frac{f(x+h) - f(x-h)}{2h}.$

A function is symmetrically differentiable at a point x if its symmetric derivative exists at that point. It can be shown that if a function is differentiable at a point, it is also symmetrically differentiable, but the converse is not true. The best known example is the absolute value function f(x) = |x|, which is not differentiable at x = 0, but is symmetrically differentiable here with symmetric derivative 0. It can also be shown that the symmetric derivative at a point is the mean of the one-sided derivatives at that point, if they both exist.