In mathematics, the symmetric derivative is an operation generalizing the ordinary derivative.
It is defined as:[1][2]
The expression under the limit is sometimes called the symmetric difference quotient.[3][4] 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 (in the usual sense) 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) = |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.[3]
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.[1][2]: 6
Neither Rolle's theorem nor the mean-value theorem hold for the symmetric derivative; some similar but weaker statements have been proved.
^ abPeter R. Mercer (2014). More Calculus of a Single Variable. Springer. p. 173. ISBN 978-1-4939-1926-0.
^ abThomson, Brian S. (1994). Symmetric Properties of Real Functions. Marcel Dekker. ISBN 0-8247-9230-0.
^ abPeter D. Lax; Maria Shea Terrell (2013). Calculus With Applications. Springer. p. 213. ISBN 978-1-4614-7946-8.
^Shirley O. Hockett; David Bock (2005). Barron's how to Prepare for the AP Calculus. Barron's Educational Series. pp. 53. ISBN 978-0-7641-2382-5.
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