Why isn't the following example a differentiable structure?

Question

Why isn't $Id_{\mathbb{R}}:(\mathbb{R},x_1) \rightarrow (\mathbb{R},x_2)$ with $x_1: x \rightarrow x$ and $x_2: x \rightarrow x^3$ a diffeomorphism?

Here $(\mathbb{R},x_1)$ and $(\mathbb{R},x_2)$ are the corresponding differentiable structures...

Answer 1

Because the overlap map $x_1 \circ x_2^{-1} : x \mapsto \sqrt[3]{x}$ is not differentiable at $0$.