\displaystyle [\partial_x f, \partial_y f, \partial_z f]
=[\partial_r f, \partial_\varphi f, \partial_\psi f] \cdot\\ \cdot\frac{1}{r\cos 2\varphi \cos\psi}\left[\begin{array}{ccc}
r\cos^2 \psi \cos\varphi & -r\cos^2 \psi \sin\varphi & r\cos 2\varphi\cos\psi\sin\psi\\
\sin\varphi & -\cos\varphi & 0\\
-\cos\varphi\cos\psi\sin\psi & \sin\varphi\cos\psi\sin\psi & \cos 2\varphi\cos^2\psi
\end{array}\right]