Namespaces
Variants
Actions

Difference between revisions of "One-sided derivative"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
−
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" />:
+
<!--
 +
o0682401.png
 +
$#A+1 = 6 n = 0
 +
$#C+1 = 6 : ~/encyclopedia/old_files/data/O068/O.0608240 One\AAhsided derivative
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
−
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068240/o0682403.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
−
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]].
+
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 $
 +
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|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
Retrieved from "https://encyclopediaofmath.org/index.php?title=One-sided_derivative&oldid=48044"