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[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.
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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

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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 5184 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)
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