I'm reading this paper. In it, the authors show this lemma:
And then they prove this lemma
My question is: I have no idea how they get the result in Lemma 3.2. Do we not get $$\frac{d}{ds}\log \lVert u(s) \rVert_{r(s)}^{r(s)} = \frac{\frac{d}{ds}\lVert u(s) \rVert_{r(s)}^{r(s)}}{\lVert u(s) \rVert_{r(s)}^{r(s)}}= \frac{\frac{d}{ds}\varphi(r(s),s)}{\lVert u(s) \rVert_{r(s)}^{r(s)}}$$ and then we can plug in Lemma 3.1. But this obviously not agree with the result in Lemma 3.2. Both terms on the RHS are lacking a factor of $\frac{1}{\lVert u(s) \rVert_{r(s)}^{r(s)}}$.