For reference, I'm following the first line in proof of Lemma 6.1 of this paper. This is my first time working with asymptotic flatness, and I would greatly appreciate it someone would tell me if the general ideas in my calculations are correct. If I haven't made any major errors, I'd appreciate if someone can help me finish the proof.
We work in an asymptotically flat 3-dimensional manifold. My question concerns the second equality of $$\nabla|\nabla u|=\nabla(g^{11})^{1/2}=-\frac{1}{2}\nabla g_{11}+O(|x|^{-1-2q}),$$ where $u=x^1$ is a function defined previously, and $x^1,x^2,x^3$ are coordinates satisfying the decay conditions $|\partial^k(g_{ij}-\delta_{ij})|=O(|x|^{-q-k})$ for some $q>1/2$ and $k=0,1,2$. For simplicitly, let me denote $O(|x|^{-p})$ by $O_{-p}$ for all decay rates $p$. For completeness, the first equality follows from $$|\nabla u|^2=g(\nabla u,\nabla u)=g(\nabla x^1,\nabla x^1)=g(dx^1,dx^1)=g^{11}.$$ From here, we take the gradient of $\delta^i_j=g^{ik}g_{kj}$. $$0=\nabla(\delta^1_1)=\nabla(g^{11}g_{11}+g^{12}{g_{12}}+g^{13}g_{13})=(\nabla g^{11})g_{11}+g^{11}(\nabla g_{11})+\underbrace{\nabla(g^{12}g_{12}+g^{13}g_{13})}_{=O_{-2q-1}},$$ The decay on the last term follows from the following calculation. By AF, both $g_{ij},g^{ij}$ can be replaced by their Euclidean counterparts up to an error of $O_{-q}$. Therefore, $$\nabla (g^{12}g_{12})=\nabla((\underbrace{\delta^{12}}_{=0}+O_{-q})(\underbrace{\delta_{12}}_{=0}+O_{-q}))=\nabla(O_{-2q})=O_{-2q-1},$$ and similarly for $g^{13}g_{13}$ thereby finishing the claim. We continue, \begin{align*} 0&=(\nabla g^{11})(\underbrace{\delta_{11}}_{=1}+O_{-q})+(\underbrace{\delta^{11}}_{=1}+O_{-q})(\nabla g_{11})+O_{-2q-1}\\ &=\nabla g^{11}+(\nabla g^{11})O_{-q}+\nabla g_{11}+O_{-q}(\nabla g_{11})+O_{-2q-1}\\ &=\nabla g^{11}+O_{-q-1} O_{-q}+\nabla g_{11}+O_{-q}O_{-q-1}+O_{-2q-1}\\ &=\nabla g^{11}+\nabla g_{11}+O_{-2q-1}. \end{align*} It follows that $\nabla g^{11}=-\nabla g_{11}+O_{-2q-1}$. From here, we calculate \begin{align*} \nabla (g^{11})^{1/2}&=\frac{1}{2}(g^{11})^{-1/2}\nabla g^{11}\\ &=\frac{1}{2}(1+O_{-q})^{-1/2}(-\nabla g_{11}+O_{-2q-1})\\ \end{align*} From here, I'm unsure how to proceed with dealing with the factor $(1+O_{-q})^{-1/2}$.