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Left and right derivative
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Left and right derivative : ウィキペディア英語版
Left and right derivative

In mathematics, the terms left derivative and right derivative are used to describe certain kinds of derivatives.
==Derivatives arising from one-sided limits==
The most common use is to describe the one-sided limit of a function defined on a subset of the real line, (see semi-differentiability). Under this notation, the right derivative is defined as
:\partial_+f(a):=\lim_\frac
and the left derivative as
:\partial_-f(a):=\lim_\frac.
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.

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