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. 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.