PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (511)
Corrections (9)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] one-sided derivatives (Definition)
  • If the real function $ f$ is defined in the point $ x_0$ and on some interval left from this and if the left-hand one-sided limit $ \lim_{h\to 0-}\frac{f(x_0+h)-f(x_0)}{h}$ exists, then this limit is defined to be the left-sided derivative of $ f$ in $ x_0$.
  • If the real function $ f$ is defined in the point $ x_0$ and on some interval right from this and if the right-hand one-sided limit $ \lim_{h\to 0+} \frac{f(x_0+h)-f(x_0)}{h}$ exists, then this limit is defined to be the right-sided derivative of $ f$ in $ x_0$.

It's apparent that if $ f$ has both the left-sided and the right-sided derivative in the point $ x_0$ and these are equal, then $ f$ is differentiable in $ x_0$ and $ f'(x_0)$ equals to these one-sided derivatives. Also inversely.

Example. The real function $ x \mapsto x\sqrt{x}$ is defined for $ x \geqq 0$ and differentiable for $ x > 0$ with $ f'(x) \equiv \frac{3}{2}\sqrt{x}$. The function also has the right derivative in 0:

$\displaystyle \lim_{h \to 0+}\frac{h\sqrt{h}- 0\sqrt{0}}{h} = \lim_{h \to 0+}\sqrt{h} = 0$

Remark. For a function $ f\!: [a,\,b] \to \mathbb{R}$, to have a right-sided derivative at $ x = a$ with value $ d$, is equivalent to saying that there is an extension $ g$ of $ f$ to some open interval containing $ [a,\,b]$ and satisfying $ g'(a) = d$. Similarly for left-sided derivatives.



"one-sided derivatives" is owned by pahio.
(view preamble)

View style:

See Also: differentiable, one-sided limit, differentiable function, one-sided continuity

Other names:  left derivative, right derivative
Also defines:  left-sided derivative, right-sided derivative

This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: open interval, extension, equivalent, function, differentiable, right, limit, one-sided limit, interval, point, real function
There are 4 references to this entry.

This is version 6 of one-sided derivatives, born on 2006-02-01, modified 2006-09-29.
Object id is 7582, canonical name is OneSidedDerivatives.
Accessed 5952 times total.

Classification:
AMS MSC: 26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)
 26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)