How to rewrite this normal derivative in a sphere as a radial derivative?

Question

I've come across the following line of working in a set of notes on the mean value theorem for harmonic functions: $$\int_{\partial B_r(x)} \frac{\partial u}{\partial \nu}\, dS = r^{n-1} \int_{\partial B_1(0)} \frac{\partial u}{\partial r}(x+ry) \, dS(y)$$ What I don't understand is how the two derivatives are converted into one another. The $r^{n-1}$ seems to come from the Jacobian determinant $(=r^n)$ and the factor of $\frac{1}{r}$ which appears in the normal vector to the ball of radius $r$. I think this should leave you with an integrand of $\nabla u \cdot (x+ry)$ but I'm not sure how this is equivalent to the integrand in the expression above. Any help is greatly appreciated.