Let $(M,g)$ be a compact Riemannian manifold without boundary. Fix a point $x\in M$ and $N\ge 2$ large. Then there exists a metric $\tilde g$, conformal to $g$ such that $$ \det \tilde g=1+O(r^N)$$ for $r\le inj(x)$ the injectivity radius at $x$, where $r=|x|$ in the $\tilde g$-normal coordinates. References: Lee, Parker - the Yamabe Problem, THM 5.1 (existence of normal conformal coordinates)
Question: Does (and how) this imply $$ \partial_r^{(i)}\log(\sqrt{\det \tilde g})=O(r^{N-i}).$$
It seems to me that this could come from the Taylor expansion of the determinant of the metric but i cannot find a proper proof. Gursky and Malchiodi in this paper http://arxiv.org/pdf/1401.3216.pdf (Lemma 2.6) use the notation $$ \det \tilde g=1+O^{(3)}(r^N)$$ and they argue $$ \partial_r \log(\sqrt{\det \tilde g})=O^{''}(r^{N-1})$$ but it is not clear to me if i can push it to higher order derivatives and yet why they use that notation since in Lee and Parker nothing like this seems to be proved.