How can I differentiate $\displaystyle\int_{B(x,r)}f(y)\ dy+r\int_{B(0,1)}\langle\nabla f(x+rz),z\rangle\ dz$ with respect to $r$ ?
For the second integral I apply product rule, first term is $\int_{B(0,1)}\langle\nabla f(x+rz),z\rangle\ dz$ and the second $r\int_{B(0,1)}\langle\Delta f(x+rz),z^2\rangle\ dz$ am I correct ?
How to differentiate the first one ?
on the top of this page I have the second derivative, $\partial_{rr}U(x,r,t)$ and now I need the third.