Symmetric Derivative

In mathematics, the symmetric derivative is an operation that generalizes the ordinary derivative, defined as the limit, as h approaches zero, of the quantity f(x+h) minus f(x-h), divided by 2h. A function is symmetrically differentiable at a point if this limit exists there; every function differentiable in the ordinary sense at a point is also symmetrically differentiable there, though the converse fails, as shown by the absolute value function, which is not ordinarily differentiable at zero but is symmetrically differentiable there with symmetric derivative zero. Where a function's ordinary left and right derivatives both exist at a point, the symmetric derivative equals their arithmetic mean; neither Rolle's theorem nor the mean value theorem holds for the symmetric derivative in general, though weaker analogues have been proved.