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$\left[X\,Y\,Z\right]$

$\left[\mathrm{\ρ}\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)\,\mathrm{\ρ}\sin \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)\,\mathrm{\ρ}\cos \left(\mathrm{\φ}\right)\right]$

$\stackrel{\text{Jacobian}}{\to }$

${\cos \left(\mathrm{\φ}\right)}^2{\sin \left(\mathrm{\θ}\right)}^2\sin \left(\mathrm{\φ}\right){\mathrm{\ρ}}^2\+{\cos \left(\mathrm{\φ}\right)}^2{\cos \left(\mathrm{\θ}\right)}^2\sin \left(\mathrm{\φ}\right){\mathrm{\ρ}}^2\+{\sin \left(\mathrm{\θ}\right)}^2{\sin \left(\mathrm{\φ}\right)}^3{\mathrm{\ρ}}^2\+{\cos \left(\mathrm{\θ}\right)}^2{\sin \left(\mathrm{\φ}\right)}^3{\mathrm{\ρ}}^2$

$\stackrel{\text{simplify}}{\=}$

$\sin \left(\mathrm{\φ}\right){\mathrm{\ρ}}^2$

Solution from first principles

$\left[\begin{array}{ccc}\frac{\partial }{\partial \mathrm{\ρ}} X& \frac{\partial }{\partial \mathrm{\φ}} X& \frac{\partial }{\partial \mathrm{\θ}} X\\ \frac{\partial }{\partial \mathrm{\ρ}} Y& \frac{\partial }{\partial \mathrm{\φ}} Y& \frac{\partial }{\partial \mathrm{\θ}} Y\\ \frac{\partial }{\partial \mathrm{\ρ}} Z& \frac{\partial }{\partial \mathrm{\φ}} Z& \frac{\partial }{\partial \mathrm{\θ}} Z\end{array}\right]$ = $\stackrel{\text{determinant}}{\to }$${\cos \left(\mathrm{\φ}\right)}^2{\sin \left(\mathrm{\θ}\right)}^2\sin \left(\mathrm{\φ}\right){\mathrm{\ρ}}^2\+{\cos \left(\mathrm{\φ}\right)}^2{\cos \left(\mathrm{\θ}\right)}^2\sin \left(\mathrm{\φ}\right){\mathrm{\ρ}}^2\+{\sin \left(\mathrm{\θ}\right)}^2{\sin \left(\mathrm{\φ}\right)}^3{\mathrm{\ρ}}^2\+{\cos \left(\mathrm{\θ}\right)}^2{\sin \left(\mathrm{\φ}\right)}^3{\mathrm{\ρ}}^2$$\stackrel{\text{simplify}}{\=}$$\sin \left(\mathrm{\φ}\right){\mathrm{\ρ}}^2$$$

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right)\:$

$\mathrm{simplify}\left(\mathrm{Jacobian}\left(\left[\mathrm{\rho } \cos \left(\mathrm{\theta }\right)\sin \left(\mathrm{\phi }\right)\,\mathrm{\rho } \sin \left(\mathrm{\theta }\right)\sin \left(\mathrm{\phi }\right)\,\mathrm{\rho } \cos \left(\mathrm{\phi }\right)\right]\,\left[\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right]\,\mathrm{output}\=\mathrm{determinant}\right)\right)$

$\sin \left(\mathrm{\φ}\right){\mathrm{\ρ}}^2$