How to calculate the Fréchet derivative of $f: \mathbb{R}^n \to \mathbb{R}:
(x_1, \dots, x_n) \mapsto \sqrt{x_{1}^{2}+ \cdots + x_{n}^{2}}$?
I dont't know hot to fin a lineat operator such that
$$\lim_{(h_1 , \ldots h_n) \to (0. \ldots 0)} \frac{f((x_1+h_1, \dots, x_n+h_n))-f((x_1, \dots, x_n))-A((h_1, \dots, x_n))}{\sqrt{h_{1}^{2}+ \cdots + h_{n}^{2}}}=0$$
Best Answer
You can also use the following: Your function F is fréchet differentiable, if there is a linear operator F'(just notation) satisfying
$F(x+h) = F(x) + hF'(x) + o(||h||)$.
Just plug in and try to get to the form of the right hand side, that's the easiest way
EDIT: An example: $F(x,y) = 3x^2 -7y^2.$
Then
$F(x+h_1,y+h_2)=3(x+h_1)^2 - 7 (y+h_2)^2 = ... = 3h_1^2 + 6h_1x - 7h_2^2 - 14h_2y + 3x^2 - 7y^2$.
Reordering gives you
$... = \underbrace{3x^2-7y^2}_{=F(x,y)} + \underbrace{6h_1x-14h_2y}_{ } +\underbrace{3h_1^2 - 7h_2^2}_{o(||h||)}$
and the 2 terms in the middle you define as F' (note that they are what you were looking for)