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''at a point  $  x _ {0} $
of a mapping  $  f:  X \rightarrow Y $
of a normed space  $  X $
into a normed space  $  Y $''

The mapping  $  h \rightarrow D ( x _ {0} , h) $
which is linear and continuous from  $  X $
into  $  Y $
and has the property that

$$ \tag{1 }
f ( x _ {0} + h)  = \ 
f ( x _ {0} ) +
D ( x _ {0} , h) +
\epsilon ( h),
$$

where

$$ 
\lim\limits _ {\| h \| \rightarrow 0 } \ 

\frac{\| \epsilon ( h) \| }{\| h \| }
  =  0.
$$

If a mapping  $  f $
admits an expansion (1) at a point  $  x _ {0} $, 
then it is said to be Fréchet differentiable, and the actual operator

$$ 
f ^ { \prime } ( x _ {0} ) h  = \ 
D ( x _ {0} , h),\ \ 
f ^ { \prime } ( x _ {0} )  \in \ 
L ( X, Y),
$$

is called the [[Fréchet derivative|Fréchet derivative]].

For a function  $  f $
in a finite number of variables, the Fréchet differential is the linear function

$$ 
h  \rightarrow \ 
\sum _ {i = 1 } ^ { n } 
\alpha _ {i} h _ {i}  = \ 
l _ {x _ {0}  } h
$$

that has the property that

$$ \tag{2 }
f ( x _ {0} + h)  = \ 
f ( x _ {0} ) +
l _ {x _ {0}  } ( h) +
o ( | h | ),
$$

where  $  | h | = ( \sum _ {i = 1 }   ^ {n} h _ {i}  ^ {2} )  ^ {1/2} $
or any other equivalent norm in  $  \mathbf R  ^ {n} $. 
Here  $  \alpha _ {i} = \partial  f / \partial  x _ {i} \mid  _ {x _ {0}  } $
are the partial derivatives of  $  f $
at  $  x _ {0} $.

Definition (2), which is now commonplace, apparently first appeared in an explicit form in the lectures of K. Weierstrass (1861, see [[#References|[1]]]). At the end of the 19th century this definition gradually came into the textbooks (see [[#References|[2]]], [[#References|[3]]] and others). But at the time when M. Fréchet began to develop infinite-dimensional analysis, the now classical definition of the differential was so far from commonplace that even Fréchet himself supposed that his definition of the differential in an infinite-dimensional space was a new concept in the finite-dimensional case too. Nowadays the term is only used in relation to infinite-dimensional mappings. See [[Gâteaux differential|Gâteaux differential]]; [[Differential|Differential]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Dugac,   "Eléments d'analyse de Karl Weierstrass" , Paris  (1972)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O. Stolz,   "Grundzüge der Differential- und Integralrechnung" , '''1''' , Teubner  (1893)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Young,   "The fundamental theorems of the differential calculus" , Cambridge Univ. Press  (1910)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Fréchet,   "Sur la notion de différentielle"  ''C.R. Acad. Sci. Paris'' , '''152'''  (1911)  pp. 845–847; 1050–1051</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M. Fréchet,   "Sur la notion de différentielle totale"  ''Nouvelles Ann. Math. Sér. 4'' , '''12'''  (1912)  pp. 385–403; 433–449</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.N. Kolmogorov,   S.V. Fomin,   "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.M. Alekseev,   V.M. Tikhomirov,   S.V. Fomin,   "Optimal control" , Consultants Bureau  (1987)  (Translated from Russian)</TD></TR></table>