Answer
$\frac{{\partial f}}{{\partial \rho }} = \sin \phi \cos \theta \frac{{\partial f}}{{\partial x}} + \sin \phi \sin \theta \frac{{\partial f}}{{\partial y}} + \cos \phi \frac{{\partial f}}{{\partial z}}$
$\frac{{\partial f}}{{\partial \theta }} = - \rho \sin \phi \sin \theta \frac{{\partial f}}{{\partial x}} + \rho \sin \phi \cos \theta \frac{{\partial f}}{{\partial y}}$
$\frac{{\partial f}}{{\partial \phi }} = \rho \cos \phi \cos \theta \frac{{\partial f}}{{\partial x}} + \rho \cos \phi \sin \theta \frac{{\partial f}}{{\partial y}} - \rho \sin \phi \frac{{\partial f}}{{\partial z}}$
Work Step by Step
Let $f = f\left( {x,y,z} \right)$. So, the partial derivatives of $f$ with respect to $\rho$, $\theta$ and $\phi$ are
$\frac{{\partial f}}{{\partial \rho }} = \frac{{\partial f}}{{\partial x}}\frac{{\partial x}}{{\partial \rho }} + \frac{{\partial f}}{{\partial y}}\frac{{\partial y}}{{\partial \rho }} + \frac{{\partial f}}{{\partial z}}\frac{{\partial z}}{{\partial \rho }}$
$\frac{{\partial f}}{{\partial \theta }} = \frac{{\partial f}}{{\partial x}}\frac{{\partial x}}{{\partial \theta }} + \frac{{\partial f}}{{\partial y}}\frac{{\partial y}}{{\partial \theta }} + \frac{{\partial f}}{{\partial z}}\frac{{\partial z}}{{\partial \theta }}$
$\frac{{\partial f}}{{\partial \phi }} = \frac{{\partial f}}{{\partial x}}\frac{{\partial x}}{{\partial \phi }} + \frac{{\partial f}}{{\partial y}}\frac{{\partial y}}{{\partial \phi }} + \frac{{\partial f}}{{\partial z}}\frac{{\partial z}}{{\partial \phi }}$
Recall the relations from spherical to rectangular coordinates given in Section 13.7:
$x = \rho \sin \phi \cos \theta $, ${\ }$ $y = \rho \sin \phi \sin \theta $, ${\ }$ $z = \rho \cos \phi $
Thus, the derivatives $\frac{{\partial f}}{{\partial \rho }}$, $\frac{{\partial f}}{{\partial \theta }}$, $\frac{{\partial f}}{{\partial \phi }}$ in terms of $\frac{{\partial f}}{{\partial x}}$, $\frac{{\partial f}}{{\partial y}}$, $\frac{{\partial f}}{{\partial z}}$ are
$\frac{{\partial f}}{{\partial \rho }} = \sin \phi \cos \theta \frac{{\partial f}}{{\partial x}} + \sin \phi \sin \theta \frac{{\partial f}}{{\partial y}} + \cos \phi \frac{{\partial f}}{{\partial z}}$
$\frac{{\partial f}}{{\partial \theta }} = - \rho \sin \phi \sin \theta \frac{{\partial f}}{{\partial x}} + \rho \sin \phi \cos \theta \frac{{\partial f}}{{\partial y}}$
$\frac{{\partial f}}{{\partial \phi }} = \rho \cos \phi \cos \theta \frac{{\partial f}}{{\partial x}} + \rho \cos \phi \sin \theta \frac{{\partial f}}{{\partial y}} - \rho \sin \phi \frac{{\partial f}}{{\partial z}}$
