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Difference between revisions of "One-sided derivative"

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A generalization of the concept of a [[Derivative|derivative]], in which the ordinary limit is replaced by a [[One-sided limit|one-sided limit]]. If the following limit exists for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068240/o0682401.png" /> of a real variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068240/o0682402.png" />:
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then it is called the right (respectively, left) derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068240/o0682404.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068240/o0682405.png" />. If the one-sided derivatives are equal, then the function has an ordinary derivative at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068240/o0682406.png" />. See also [[Differential calculus|Differential calculus]].
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A generalization of the concept of a [[Derivative|derivative]], in which the ordinary limit is replaced by a [[One-sided limit|one-sided limit]]. If the following limit exists for a function  $  f $
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of a real variable  $  x $:
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$$
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\lim\limits _ {x \rightarrow x _ {0} + 0 } \
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\frac{f ( x) - f ( x _ {0} ) }{x - x _ {0} }
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\ \
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\left ( \textrm{ or } \
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\lim\limits _ {x \rightarrow x _ {0} - 0 } \
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\frac{f ( x) - f ( x _ {0} ) }{x - x _ {0} }
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\right ) ,
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$$
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then it is called the right (respectively, left) derivative of $  f $
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at the point $  x _ {0} $.  
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If the one-sided derivatives are equal, then the function has an ordinary derivative at $  x _ {0} $.  
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See also [[Differential calculus|Differential calculus]].

Latest revision as of 08:04, 6 June 2020


A generalization of the concept of a derivative, in which the ordinary limit is replaced by a one-sided limit. If the following limit exists for a function $ f $ of a real variable $ x $:

$$ \lim\limits _ {x \rightarrow x _ {0} + 0 } \ \frac{f ( x) - f ( x _ {0} ) }{x - x _ {0} } \ \ \left ( \textrm{ or } \ \lim\limits _ {x \rightarrow x _ {0} - 0 } \ \frac{f ( x) - f ( x _ {0} ) }{x - x _ {0} } \right ) , $$

then it is called the right (respectively, left) derivative of $ f $ at the point $ x _ {0} $. If the one-sided derivatives are equal, then the function has an ordinary derivative at $ x _ {0} $. See also Differential calculus.

How to Cite This Entry:
One-sided derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=One-sided_derivative&oldid=48044
This article was adapted from an original article by G.P. Tolstov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article