Differentials of a multivariable functions?

Questions
1

Let’s say I have:

x=r\sin\theta\cos\phi
y=r\sin\theta\sin\phi
z=r\cos\theta

I would like to get an expression for:

dx^2+dy^2+dz^2

in terms of r, \theta, \phi.
For example:

dx=\sin\theta\cos\phi dr+r\sin\theta(-\sin\phi)d\phi+r\cos\theta\cos\phi

How can I achieve this with Mathematica?
I know I can start with:

x[_r,_theta,_phi]=r*Sin[theta]*cos[Phi]

and so on, but then I don’t see a straightforward line forward.

2

Here is a way:

{dx, dy, dz} = Grad[FromSphericalCoordinates[{r, \[Theta], \[Phi]}],
   {r, \[Theta], \[Phi]}] . {dr, d\[Theta], d\[Phi]}
dx^2 + dy^2 + dz^2