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''of a function  $  f $
at a point  $  x _ {0} $''

The value

$$ 
D  ^ {2} f( x _ {0} )  =  \lim\limits _ {h \rightarrow 0 }  
\frac{f( x _ {0} + h)- 2f( x _ {0} )+ f( x _ {0} - h) }{h  ^ {2} }
 .
$$

It is sometimes called the [[Riemann derivative|Riemann derivative]] or the second symmetric derivative. For the first time introduced by B. Riemann in 1854 (see [[#References|[2]]]); it was studied by H.A. Schwarz [[#References|[1]]]. More generally, the symmetric derivative of order  $  n $
is also called a Schwarz symmetric derivative:

$$ 
D  ^ {n} f( x)  =  \lim\limits _ {h \rightarrow 0 }  
\frac{\Delta _ {h}  ^ {n} f( x) }{h  ^ {n} }
 =
$$

$$ 
= \ 
\lim\limits _ {h \rightarrow 0 }  
\frac{\sum _ { k= } 0 ^ { n }  \left
( \begin{array}{c}
n \\
 k 
\end{array}
 \right ) (- 1)  ^ {k} f ( x + ( n- 2k ) h / 2 ) }{h  ^ {n} }
 .
$$

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H.A. Schwarz,   "Beweis eines für die Theorie der trigonometrischen Reihen in Betracht kommenden Hülfssatzes" , ''Gesammelte Math. Abhandlungen'' , Chelsea, reprint  (1972)  pp. 341–343</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B. Riemann,   "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe"  H. Weber (ed.) , ''B. Riemann's Gesammelte Mathematische Werke'' , Dover, reprint  (1953)  pp. 227–271  ((Original: Göttinger Akad. Abh. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083580/s0835808.png" /> (1868)))</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.P. Natanson,   "Theory of functions of a real variable" , '''1–2''' , F. Ungar  (1955–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,   "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>

====Comments====
The name general derivative is also used for this notion. A natural approach is to start with the central difference  $  f( x _ {0} + h/2 ) - f( x _ {0} - h/2) $, 
and to define the first symmetric derivative as

$$ 
Df( x _ {0} )  =  \lim\limits _ {h \rightarrow 0 } \ 

\frac{f( x _ {0} + h/2)- f( x _ {0} - h/2) }{h}
  = \ 
\lim\limits _ {h \rightarrow 0 } \ 

\frac{\Delta _ {h} f ( x _ {0} ) }{h}
 ,
$$

and then  $  D  ^ {n} = D( D  ^ {n-} 1 ) $, 
$  n \geq  1 $, 
$  D  ^ {0} f = f $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,   "Real and complex analysis" , McGraw-Hill  (1974)  pp. 24</TD></TR></table>