# one-sided derivatives

• 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 $f^{\prime}(x_{0})$ 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  $f^{\prime}(x)\equiv\frac{3}{2}\sqrt{x}$.  The function also has the right derivative in $0$:

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

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  $g^{\prime}(a)=d$.  Similarly for left-sided derivatives.

 Title one-sided derivatives Canonical name OnesidedDerivatives Date of creation 2013-03-22 15:39:00 Last modified on 2013-03-22 15:39:00 Owner pahio (2872) Last modified by pahio (2872) Numerical id 9 Author pahio (2872) Entry type Definition Classification msc 26B05 Classification msc 26A24 Synonym left derivative Synonym right derivative Related topic Differentiable Related topic OneSidedLimit Related topic DifferntiableFunction Related topic OneSidedContinuity Related topic SemicubicalParabola Defines left-sided derivative Defines right-sided derivative