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In calculus, the notions of one sided differentiability and semi differentiability of a real valued function f of a real variable are weaker than differentiability. Specifically, the function f is said to be right differentiable at a point a if, roughly speaking, a derivative can be defined as the function's argument x moves to a from the right, and left differentiable at a if the derivative can be defined as x moves to a from the left.

In mathematics, a left derivative and a right derivative are derivatives (rates of change of a function) defined for movement in one direction only (left or right; that is, to lower or higher values) by the argument of a function.

A function is differentiable at an interior point a of its domain if and only if it is semi differentiable at a and the left derivative is equal to the right derivative. An example of a semi differentiable function, which is not differentiable, is the absolute value function , at a = 0. We find easily
If a function is semi differentiable at a point a, it implies that it is continuous at a. The indicator function 1[0,∞) is right differentiable at every real a, but discontinuous at zero (note that this indicator function is not left differentiable at zero).

If a real valued, differentiable function f, defined on an interval I of the real line, has zero derivative everywhere, then it is constant, as an application of the mean value theorem shows. The assumption of differentiability can be weakened to continuity and one sided differentiability of f. The version for right differentiable functions is given below, the version for left differentiable functions is analogous.

Another common use is to describe derivatives treated as binary operators in infix notation, in which the derivatives is to be applied either to the left or right operands. This is useful, for example, when defining generalizations of the Poisson bracket. For a pair of functions f and g, the left and right derivatives are respectively defined as

In bra–ket notation, the derivative operator can act on the right operand as the regular derivative or on the left as the negative derivative.

Any convex function on a convex open subset of Rn is semi differentiable. While every semi differentiable function of one variable is continuous; this is no longer true for several variables.

Instead of real valued functions, one can consider functions taking values in Rn or in a Banach space.