Solution
= Solution
For $r>0$, the <partial derivatives> of radius are $r_x=x/r$, $r_y=y/r$ and $r_z=z/r$. Applying the <chain rule> to the outer <sine> and then the radial profile gives the <Cartesian coordinates> of the <gradient>:
$$
\boxed{\nabla\sin[f(r)]=\frac{f'(r)\cos[f(r)]}{r}(x,y,z).}
$$
At the origin these coordinate formulas require a differentiable radial extension; differentiability of $f$ as a one-variable profile alone does not guarantee it.