Symmetric derivative

File:Modulusfunction.png of the absolute value function. Note the sharp turn at {{math|1=x = 0}}, leading to non-differentiability of the curve at {{math|1=x = 0}}. The function hence possesses no ordinary derivative at {{math|1=x = 0}}. The symmetric derivative, however, exists for the function at {{math|1=x = 0}}.]]

For the absolute value function $f(x)\; =\; |x|$, using the notation $f\_s(x)$ for the symmetric derivative, we have at $x\; =\; 0$ that

$$\backslash begin\{align\}$$

f_s(0)

&= \lim_{h \to 0}\frac{f(0 + h) - f(0 - h)}{2h} = \lim_{h \to 0}\frac{f(h) - f(-h)}{2h} \\

&= \lim_{h \to 0}\frac

&= \lim_{h \to 0}\frac

&= \lim_{h \to 0}\frac{0}{2h} = 0. \\

\end{align}

Hence the symmetric derivative of the absolute value function exists at $x\; =\; 0$ and is equal to zero, even though its ordinary derivative does not exist at that point (due to a "sharp" turn in the curve at $x\; =\; 0$).

Note that in this example both the left and right derivatives at 0 exist, but they are unequal (one is −1, while the other is +1); their average is 0, as expected.

=== The function x⁻² ===

File:Graphinversesqrt.png

For the function $f(x)\; =\; 1/x^2$, at $x\; =\; 0$ we have

$$\backslash begin\{align\}$$

f_s(0)

&= \lim_{h \to 0}\frac{f(0 + h) - f(0 - h)}{2h}

= \lim_{h \to 0}\frac{f(h) - f(-h)}{2h} \\[1ex]

&= \lim_{h \to 0}\frac{1/h^2 - 1/(-h)^2}{2h}

= \lim_{h \to 0}\frac{1/h^2 - 1/h^2}{2h}

= \lim_{h \to 0}\frac{0}{2h} = 0.

\end{align}

Again, for this function the symmetric derivative exists at $x\; =\; 0$, while its ordinary derivative does not exist at $x\; =\; 0$ due to discontinuity in the curve there. Furthermore, neither the left nor the right derivative is finite at 0, i.e. this is an essential discontinuity.

The Dirichlet function, defined as:

$$f(x)\; =\; \backslash begin\{cases\}$$

1, & \text{if }x\text{ is rational} \\

0, & \text{if }x\text{ is irrational}

\end{cases}

has a symmetric derivative at every $x\; \backslash in\; \backslash Q$, but is not symmetrically differentiable at any $x\; \backslash in\; \backslash R\; \backslash setminus\; \backslash Q$; i.e. the symmetric derivative exists at rational numbers but not at irrational numbers.

The symmetric derivative does not obey the usual mean-value theorem (of Lagrange). As a counterexample, the symmetric derivative of {{math|1=f(x) = {{!}}x{{!}}}} has the image {{math|{−1, 0, 1}}}, but secants for f can have a wider range of slopes; for instance, on the interval {{closed-closed|−1, 2}}, the mean-value theorem would mandate that there exist a point where the (symmetric) derivative takes the value {{nowrap|$\backslash frac$

{{expand section|date=April 2015}}

The notion generalizes to higher-order symmetric derivatives and also to n-dimensional Euclidean spaces.

The second symmetric derivative is defined as{{rp|p=1}}

$$\backslash lim\_\{h\; \backslash to\; 0\}\; \backslash frac\{f(x\; +\; h)\; -\; 2f(x)\; +\; f(x\; -\; h)\}\{h^2\}.$$

If the (usual) second derivative exists, then the second symmetric derivative exists and is equal to it.{{cite book |author=A. Zygmund |title=Trigonometric Series |title-link= Trigonometric Series |year=2002 |publisher=Cambridge University Press |isbn=978-0-521-89053-3 |pages=22–23}} The second symmetric derivative may exist, however, even when the (ordinary) second derivative does not. As example, consider the sign function $\backslash sgn(x)$, which is defined by

$$\backslash sgn(x)\; =\; \backslash begin\{cases\}$$

-1 & \text{if } x < 0, \\

0 & \text{if } x = 0, \\

1 & \text{if } x > 0.

\end{cases}

The sign function is not continuous at zero, and therefore the second derivative for $x\; =\; 0$ does not exist. But the second symmetric derivative exists for $x\; =\; 0$:

$$\backslash lim\_\{h\; \backslash to\; 0\}\; \backslash frac\{\backslash sgn(0\; +\; h)\; -\; 2\backslash sgn(0)\; +\; \backslash sgn(0\; -\; h)\}\{h^2\}\; =\; \backslash lim\_\{h\; \backslash to\; 0\}\; \backslash frac\{\backslash sgn(h)\; -\; 2\backslash cdot\; 0\; +\; (-\backslash sgn(h))\}\{h^2\}\; =\; \backslash lim\_\{h\; \backslash to\; 0\}\; \backslash frac\{0\}\{h^2\}\; =\; 0.$$

• Central differencing scheme
• Density point
• Generalizations of the derivative
• Symmetrically continuous function

{{reflist}}

• {{cite book |author=A. B. Kharazishvili |title=Strange Functions in Real Analysis |edition=2nd |year=2005 |publisher=CRC Press |isbn=978-1-4200-3484-4 |page=34}}
• {{cite journal |author=Aull, C. E. |title=The first symmetric derivative |journal=Am. Math. Mon. |volume=74 |issue=6 |pages=708–711 |year=1967 |doi=10.1080/00029890.1967.12000020}}

• {{springer|title=Symmetric derivative|id=p/s091610}}
• [http://demonstrations.wolfram.com/ApproximatingTheDerivativeByTheSymmetricDifferenceQuotient/ Approximating the Derivative by the Symmetric Difference Quotient (Wolfram Demonstrations Project)]

Category:Differential calculus

Category:Generalizations of the derivative