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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 to be symmetrically differentiable at a point x if its symmetric derivative exists at that point.
If a function is differentiable at a point, then it is also symmetrically differentiable, but the converse is not true. A well-known counterexample is the absolute value function f = |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.