Frechet Derivative of a direct product of functions

Question

Given two functions $f: U \to \mathbb{Y}$ and $g: U \to \mathbb{Y}$ (where $U\subset \mathbb{X}$ is open) that are Frechet differentiable at $x$. Also, $||(x,y)||_{\mathbb{X} \times \mathbb{X}}=||x||_{\mathbb{X}} + ||y||_{\mathbb{X}}$. What is the Frechet derivate of $f \times g: U \times U \to \mathbb{Y} \times \mathbb{Y}$ at $x$, where $(f \times g)(u,v)=(f(u),g(v))$

I have proven that $f \times g$ is Frechet differentiable at $x$, but I cannot find an appropriate derivative for it.

Answer 1

$D(f\times g)(u,v)=\begin{pmatrix}Df(u) &0\\ 0&Dg(v)\end{pmatrix}$