Is $x^{1/3}$ differentiable at $0$?

The function $x^{1/3}$ is not differentiable at $x=0$, but the graph $\{(x,x^{1/3})\colon x\in\mathbb{R}\}\subseteq \mathbb{R}^2$ is a smooth submanifold, something that for example does not happen with $x^{2/3}$. I believe this latter notion is the one that reconciles your discrepancy.