Differentiating measure of a submanifold depending on a parameter

Question

I've been studying the following variational problem. Among all codimension 1 submanifolds of Euclidean space enclosing a given volume, does the sphere minimize the average distance from the submanifold to the origin? In particular I'm dealing with submanifolds of the form $$\{x: u(x)=t\}\ ,$$ where t is a fixed parameter and u is smooth with compact support. Defining the functional $$A_{t}(u):=\frac{1}{H_{n-1}(\{x: u(x)=t\})}\int_{u(x)=t}|x|dH_{n-1},$$ I'd like to show that $$ A_{t}(u) \geq A_{t}(u^*),$$ where $u^*$ denotes the decreasing spherical symmetrization of u (whose level sets are indeed spheres). In deriving the Euler-Lagrange equation of this functional, I'm faced with expressions like $$\frac{d}{d\epsilon}|_{\epsilon=0} H_{n-1}\{x:u(x)+\epsilon\varphi(x)=t\} $$ and $$ \frac{d}{d\epsilon}|_{\epsilon=0} \int_{(u+\epsilon\varphi)(x)=t}|x|dH_{n-1}.$$ I was wondering if these expressions could be differentiated in closed form to verify that a sphere is indeed a minimizer. Thanks.