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In [[mathematics]], the '''symmetric derivative''' is an [[Operator (mathematics)|operation]] related to the ordinary [[derivative]]. | |||
It is defined as: | |||
:<math>\lim_{h \to 0}\frac{f(x+h) - f(x-h)}{2h}.</math> | |||
A function is '''symmetrically differentiable''' at a point ''x'' if its symmetric derivative exists at that point. It can be shown that if a function is [[differentiable function|differentiable]] at a point, it is also symmetrically differentiable, but the converse is not true. The best known example is the [[absolute value]] function f(x) = |x|, which is not differentiable at x = 0, but is symmetrically differentiable here with symmetric derivative 0. It can also be shown that the symmetric derivative at a point is the mean of the one-sided derivatives at that point, if they both exist. | |||
==Examples== | |||
1. The [[modulus function]],<math>f(x)= \left\vert x \right\vert</math> <br /> | |||
For [[absolute value function]], or the [[modulus function]], we have, at <math>x=0</math>, | |||
:<math>\begin{matrix} | |||
\\ f_s(0)= \lim_{h \to 0}\frac{f(0+h) - f(0-h)}{2h} \\ | |||
\\ f_s(0)= \lim_{h \to 0}\frac{f(h) - f(-h)}{2h} \\ | |||
\\ f_s(0)= \lim_{h \to 0}\frac{\left\vert h \right\vert - \left\vert -h \right\vert}{2h} \\ | |||
\\ f_s(0)= \lim_{h \to 0}\frac{h-(-(-h))}{2h} \\ | |||
\\ f_s(0)= 0 \\ | |||
\end{matrix}</math> | |||
only, where remember that <math> h>0 </math> and <math> h\longrightarrow 0</math>, and hence <math>\left\vert -h \right\vert</math> is equal to <math>-(-h)</math> only! So, we observe that the symmetric derivative of the modulus function exists at <math>x=0</math>,and is equal to zero, even if its ordinary derivative won't exist at that point (due to a "sharp" turn in the curve at <math>x=0</math>). | |||
[[File:Modulusfunction.png|thumb|center|Graph of the [[Modulus Function]] y=|x|. Note the sharp turn at x=0, leading to non differentiability of the curve at x=0. The function hence possesses no ordinary derivative at x=0. Symmetric Derivative, however exists for the function at x=0.]] | |||
2. The function <math> f(x)=1/x^2</math> <br /> | |||
For the function <math> f(x)=1/x^2</math>, we have, at <math>x=0</math>, | |||
:<math>\begin{matrix} | |||
\\ f_s(0)= \lim_{h \to 0}\frac{f(0+h) - f(0-h)}{2h} \\ | |||
\\ f_s(0)= \lim_{h \to 0}\frac{f(h) - f(-h)}{2h} \\ | |||
\\ f_s(0)= \lim_{h \to 0}\frac{1/h^2 - 1/(-h)^2}{2h} \\ | |||
\\ f_s(0)= \lim_{h \to 0}\frac{1/h^2-1/h^2}{2h} \\ | |||
\\ f_s(0)= 0 \\ | |||
\end{matrix}</math> | |||
only, where again, <math> h>0 </math> and <math> h\longrightarrow 0</math>. See that again, for this function, its symmetric derivative exists at <math>x=0</math>, its ordinary derivative does not occur at <math>x=0</math>, due to discontinuity in the curve at <math>x=0</math> (i.e. essential discontinuity). | |||
[[File:Graphinversesqrt.png|thumb|center|Graph of y=1/x². Note the discontinuity at x=0. The function hence possesses no ordinary derivative at x=0. Symmetric Derivative, however exists for the function at x=0.]] | |||
3. The [[Dirichlet function]], defined as: | |||
<math>f(x) = | |||
\begin{cases} | |||
1, & \text{if }x\text{ is rational} \\ | |||
0, & \text{if }x\text{ is irrational} | |||
\end{cases}</math> | |||
may be analysed to realize that it has symmetric derivatives <math> \forall x \in \mathbb{Q}</math> but not <math>\forall x \in \mathbb{R}-\mathbb{Q}</math>, i.e. symmetric derivative exists for rational numbers bur not for irrational numbers. | |||
== See also == | |||
* [[Symmetrically continuous function]] | |||
== References == | |||
* {{cite book | |||
| first= Brian S. | |||
| last= Thomson | |||
| year= 1994 | |||
| title= Symmetric Properties of Real Functions | |||
| publisher= Marcel Dekker | |||
| isbn= 0-8247-9230-0 | |||
}} | |||
==External links== | |||
*[http://demonstrations.wolfram.com/ApproximatingTheDerivativeByTheSymmetricDifferenceQuotient/ Approximating the Derivative by the Symmetric Difference Quotient (Wolfram Demonstrations Project)] | |||
*[http://mathworld.wolfram.com/DirichletFunction.html Dirichlet Function] | |||
*[http://math.feld.cvut.cz/mt/txtb/4/txe3ba4s.htm Dirichlet function and its modifications] | |||
*[http://www.wolframalpha.com/input/?i=dirichlet+function&a=ClashPrefs_*MathWorld.DirichletFunction- Dirichlet function-Wolfram Alpha] | |||
[[Category:Differential calculus]] |
Revision as of 15:28, 23 January 2014
In mathematics, the symmetric derivative is an operation related to the ordinary derivative.
It is defined as:
A function is symmetrically differentiable at a point x if its symmetric derivative exists at that point. It can be shown that if a function is differentiable at a point, it is also symmetrically differentiable, but the converse is not true. The best known example is the absolute value function f(x) = |x|, which is not differentiable at x = 0, but is symmetrically differentiable here with symmetric derivative 0. It can also be shown that the symmetric derivative at a point is the mean of the one-sided derivatives at that point, if they both exist.
Examples
1. The modulus function,
For absolute value function, or the modulus function, we have, at ,
only, where remember that and , and hence is equal to only! So, we observe that the symmetric derivative of the modulus function exists at ,and is equal to zero, even if its ordinary derivative won't exist at that point (due to a "sharp" turn in the curve at ).
2. The function
For the function , we have, at ,
only, where again, and . See that again, for this function, its symmetric derivative exists at , its ordinary derivative does not occur at , due to discontinuity in the curve at (i.e. essential discontinuity).
3. The Dirichlet function, defined as:
may be analysed to realize that it has symmetric derivatives but not , i.e. symmetric derivative exists for rational numbers bur not for irrational numbers.
See also
References
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