Symmetric difference quotient

In mathematics, the symmetric derivative is an operation generalizing the ordinary derivative. It is defined as:

The expression under the limit is sometimes called the symmetric difference quotient. A function is said symmetrically differentiable at a point x if its symmetric derivative exists at that point.

If a function is differentiable (in the usual sense) at a point, then it is also symmetrically differentiable, but the converse is not true. A well-known [counter]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. For differentiable functions, the symmetric difference quotient does provide a better numerical approximation of the derivative than the usual difference quotient.

The symmetric derivative at a given point equals the arithmetic mean of the left and right derivatives at that point, if the latter two both exist.

Neither Rolle's theorem nor the mean value theorem hold for the symmetric derivative; some similar but weaker statements have been proved.

For the modulus function, ${\displaystyle f(x)=\left\vert x\right\vert }$, we have, at ${\displaystyle x=0}$,

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