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A weak extension of the operation of ordinary [[Differentiation|differentiation]]. Let  $  f $
be a [[Generalized function|generalized function]],  $  f \in D  ^  \prime  ( O) $. 
The generalized (weak) derivative

$$ 
D  ^  \alpha  f  = \ 

\frac{\partial  ^ {| \alpha | } f }{\partial  x _ {1} ^ {\alpha _ {1} } \dots
\partial  x _ {n} ^ {\alpha _ {n} } }
 ,\ \ 
| \alpha | = \alpha _ {1} + \dots + \alpha _ {n} ,
$$

of order  $  \alpha = ( \alpha _ {1} \dots \alpha _ {n} ) $
is defined by the equation

$$ \tag{* }
( D  ^  \alpha  f , \phi )  = \ 
( - 1 ) ^ {| \alpha | }
( f , D  ^  \alpha  \phi ) ,\ \ 
\phi \in D ( O) .
$$

Since the operation  $  \phi \mapsto (- 1) ^ {| \alpha | } D  ^  \alpha  \phi $
is linear and continuous from  $  D ( O) $
into  $  D ( O) $, 
the functional  $  D  ^  \alpha  f $
defined by the right-hand side of (*) is a generalized function in  $  D  ^  \prime  ( O) $. 
If  $  f \in C  ^ {p} ( O) $, 
then  $  D  ^  \alpha  f \in C ^ {p - | \alpha | } ( O) $
for all  $  \alpha $
with  $  | \alpha | \leq  p $.

The following properties hold for the derivatives of a generalized function: the operation  $  f \mapsto D  ^  \alpha  f $
is linear and continuous from  $  D  ^  \prime  ( O) $
into  $  D  ^  \prime  ( O) $; 
any generalized function in  $  D  ^  \prime  ( O) $
is infinitely differentiable (in the generalized sense); the result of differentiation does not depend on the order; the [[Leibniz formula|Leibniz formula]] is valid for the differentiation of a product  $  af $, 
when  $  a \in C  ^  \infty  ( O) $; 
and  $  \supp  D  ^  \alpha  f \subset  \supp  f $.

Let  $  f \in L _ { \mathop{\rm loc}  }  ^ {1} ( O) $. 
It may happen that a certain generalized derivative can be identified with some  $  L _ { \mathop{\rm loc}  }  ^ {1} ( O) $-
function. In this case  $  D  ^  \alpha  f ( x) $
is a [[Generalized derivative|generalized derivative]] of function type.

===Examples.===

1)  $  \theta  ^  \prime  = \delta $, 
where  $  \theta $
is the Heaviside function and  $  \delta $
is the Dirac function (cf. [[Delta-function|Delta-function]] for both).

2) The general solution of the equation  $  u  ^  \prime  = 0 $
in the class  $  D  ^  \prime  $
is an arbitrary constant.

3) The [[Trigonometric series|trigonometric series]]

$$ 
\sum _ {k = - \infty } ^  \infty   a _ {k} e  ^ {ikx} ,\ \ 
| a _ {k} |  \leq   A ( 1 + | k | )  ^ {m} ,
$$

converges in  $  D  ^  \prime  $
and it can be differentiated term-by-term in  $  D  ^  \prime  $
infinitely many times.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Schwartz,   "Théorie des distributions" , '''1''' , Hermann  (1950)  {{MR|0035918}} {{ZBL|0037.07301}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.L. Sobolev,   "Applications of functional analysis in mathematical physics" , Amer. Math. Soc.  (1963)  (Translated from Russian)  {{MR|0165337}} {{ZBL|0123.09003}} </TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Yosida,   "Functional analysis" , Springer  (1980)  pp. Chapt. 8, Sect. 4; 5  {{MR|0617913}} {{ZBL|0435.46002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)  {{MR|0717035}} {{MR|0705278}} {{ZBL|0521.35002}} {{ZBL|0521.35001}} </TD></TR></table>