consider function from open set $U = \{\textbf{x} : \textbf{x} \in \mathbb{R}^4 , x_1 < x_2<x_3<x_4 , \} \subset \mathbb{R}^4$ to $\mathbb{R}$. I want to show that function is differetiable.
$$f\left(\begin{matrix} x_1\\x_2 \\x_3 \\ x_4\end{matrix}\right) = \log(x_3 - x_1) + \log (x_4 -x_2)$$
I will write $f$ as composition of three functions
1.$h\left(\begin{matrix} x_1\\x_2 \\x_3 \\ x_4\end{matrix}\right) = \left( \begin{matrix} x_3 - x_1 \\ x_4-x_2 \end{matrix}\right)$ and $h$ is differentiable because it is co-ordinate wise differentiable
2.$g\left(\begin{matrix} x_1\\x_2 \end{matrix}\right)= \left(\begin{matrix} \log x_1\\ \log x_2 \end{matrix}\right) $ $g$ is differentiable because it is co-ordinate wise differentiable on $\{ \textbf{x} \in \mathbb{R}^2: x_1> 0,x_2 > 0\}$
3.$k\left(\begin{matrix} x_1\\x_2 \end{matrix}\right)= x_1 +x_2 $, $k$ is obviously differentiable,
and $$f(\textbf{x}) = (k\circ g\circ h )(\textbf{x})$$
Since each of these function are differentiable over relevant domains.We can conclude that function $f$ is differentiable.Is there any flaw in my argument?