there. However, it has a right derivative at all points, with \partial_+f(a) constantly equal to 0. 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.
Definitions Let
f denote a real-valued function defined on a subset
I of the real numbers. If is a
limit point of and the
one-sided limit :\partial_+f(a):=\lim_\frac{f(x)-f(a)}{x-a} exists as a real number, then
f is called
right differentiable at
a and the limit
∂+
f(
a) is called the
right derivative of
f at
a. If is a limit point of and the one-sided limit :\partial_-f(a):=\lim_\frac{f(x)-f(a)}{x-a} exists as a real number, then
f is called
left differentiable at
a and the limit
∂–
f(
a) is called the
left derivative of
f at
a. If is a limit point of and and if
f is left and right differentiable at
a, then
f is called
semi-differentiable at
a. If the left and right derivatives are equal, then they have the same value as the usual ("bidirectional") derivative. One can also define a
symmetric derivative, which equals the
arithmetic mean of the left and right derivatives (when they both exist), so the symmetric derivative may exist when the usual derivative does not.
Remarks and examples • 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 f(x)=|x| , at
a = 0. We find easily \partial_-f(0)=-1, \partial_+f(0)=1. • 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).
Application 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. {{math proof| For a
proof by contradiction, assume there exist in
I such that . Then :\varepsilon:=\frac{2(b-a)}>0. Define
c as the
infimum of all those
x in the interval for which the
difference quotient of
f exceeds
ε in absolute value, i.e. :c=\inf\{\,x\in(a,b]\mid |f(x)-f(a)|>\varepsilon(x-a)\,\}. Due to the continuity of
f, it follows that and . At
c the right derivative of
f is zero by assumption, hence there exists
d in the interval with for all
x in . Hence, by the
triangle inequality, :|f(x)-f(a)|\le|f(x)-f(c)|+|f(c)-f(a)|\le\varepsilon(x-a) for all
x in , which contradicts the definition of
c.}}
Differential operators acting to the left or the right 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 :f \stackrel{\leftarrow }{\partial_x} g = \frac{\partial f}{\partial x} \cdot g :f \stackrel{\rightarrow }{\partial_x} g = f \cdot \frac{\partial g}{\partial x}. 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. ==Higher-dimensional case==