How to calculate a Fréchet derivative?

Question

What is the standard algorithm for calculating a Fréchet derivative? i.e.

$f(x,y)=x^2y$

for $(x_0,y_0)\in\mathbb{R}^2$

Answer 1

Let $f:\mathbb R^n \to \mathbb R^m$ be (Frechet) differentiable at $x \in \mathbb R^n$. The Frechet derivative of $f$ at $x$, which is a linear transformation from $\mathbb R^n$ to $\mathbb R^m$, is represented (with respect to the standard bases of $\mathbb R^n$ and $\mathbb R^m$) by the matrix \begin{equation*} f'(x) = \begin{bmatrix} \frac{\partial f_1(x)}{\partial x_1} & \cdots & \frac{\partial f_1(x)}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m(x)}{\partial x_1} & \cdots & \frac{\partial f_m(x)}{\partial x_n} \end{bmatrix}. \end{equation*} Here $f_i$ is the $i$th component function of $f$. In the example you gave, you can compute these partial derivatives explicitly.