= Solution
The three <coordinate basis> vectors are the partial <derivatives> of the position <vector> with respect to the three coordinates. Direct differentiation gives
$$
\boxed{\mathbf h_\xi=\frac{\partial\mathbf x}{\partial\xi}=\begin{pmatrix}\cosh\xi\sin\theta\cos\phi\\\cosh\xi\sin\theta\sin\phi\\g'(\xi)\cos\theta\end{pmatrix},\qquad\mathbf h_\theta=\frac{\partial\mathbf x}{\partial\theta}=\begin{pmatrix}\sinh\xi\cos\theta\cos\phi\\\sinh\xi\cos\theta\sin\phi\\-g(\xi)\sin\theta\end{pmatrix},}
$$
$$
\boxed{\mathbf h_\phi=\frac{\partial\mathbf x}{\partial\phi}=\begin{pmatrix}-\sinh\xi\sin\theta\sin\phi\\\sinh\xi\sin\theta\cos\phi\\0\end{pmatrix}.}
$$
The multivariable <chain rule> then gives $d\mathbf x=\mathbf h_\xi d\xi+\mathbf h_\theta d\theta+\mathbf h_\phi d\phi$. These are coordinate <tangent vectors>, not necessarily unit <vectors>.
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