Suppose $f$ is a conformal map between real inner product spaces. Then for all vectors $x, v, u$, $\langle d_xfv, d_x f u\rangle = \lambda(x)\langle v, u\rangle$ for some positive $\lambda(x)$.
I am currently reading the proof of Liouville's theorem on conformal maps between real inner product spaces due to Rolf Nevanlinna. The first step is to differentiate the inner product $\langle d_xfv, d_x f u\rangle$ for some vectors $v, u$. The proof uses the fact that $\lambda$ is differentiable wrt $x$. But I don't quite understand why this is the case. Fix some $v \ne 0$, then $\lambda(x)=|d_xfv|/|v|$ for all $x$. How can I proceed from here?