# derivative of the integral over a sphere of variable radius [Reference needed]

On a Riemannian manifold $(M,g)$, let $F(s)=\int_{\partial B_s(x_0)}udS$ where

• $u$ is a smooth function,
• ${\partial B_s(x_0)}$ is the (geodesical) sphere of center $x_0$ and radius $r$
• dS is the standard (hyper)surface measure (induced by $g$ on submanifold)

My question is how to explicitly compute, for small $s$, the derivative $F^\prime(s)$.

I would really appreciate if someone can point to me a precise reference.