Leibniz formula for integral with spherical coordinates in multi-dimensional space

Question

Given $u: \mathbb{R}^n \to \mathbb{R}$ smooth enough and set $\bar{u}(r) = \int_{|x| = r}u(x)d\sigma$, with $x = \left(r, \sigma\right)$ is the spherical coordinate of $x$.
I have some differences in calculating $\frac{d^2}{dr^2} \bar{u}$.
From the Leibniz rule, we can have it for a ball (Leibniz rule with balls), but I don't know how to apply it for the spherical coordinate.
Thank you very much in advance for your help.