In a proof I want to compute the laplacian of a function in spherical coordinates. We know that : $$|x|^2\Delta=(r\partial_r)^2+(n-2)r\partial_r+\Delta_{\mathcal{S}^{n-1}}.$$
For a function $f$, I want to take the laplacian of : $f(|x|)=f(r)$.
By using the formula we obtain that : $$r^2\Delta f(r)=r^2f''(r)+(n-1)rf'(r)+\Delta_{\mathcal{S}^{n-1}}f(r).$$
Now I am stuck with $\Delta_{\mathcal{S}^{n-1}}f(r)$, I don't know how to compute it.
Thank you for you help .