Proving the generalized Riccati Equation for the second fundamental form

Question

Let $(M,g)$ be an $n$-dimensional Riemannian manifold and $u \in C^\infty(M)$ so that whenever $|\nabla u| \not = 0$, we may write the metric , by the generalized Gauss' Lemma, as $$ g = \frac{1}{|\nabla u|^2}du \otimes du + g_{ij}(u, \theta)d\theta^i \otimes d\theta^j$$ around the hypersurface $\Sigma = \{u = 0\}$. We denote by $h$ the second fundamental form of $\Sigma$ wrt the unit normal vector given by considering the $(1,0)$ version of the 1-form $\nu = \frac{\nabla u}{|\nabla u|}$. I want to prove the following equality: $$ \frac{\partial h_{ij}}{\partial u} = \frac{h_{ij} + R_{i \nu j \nu}}{|\nabla u|} -\nabla^{\Sigma}_i \nabla^{\Sigma}_j(\frac{1}{|\nabla u|})$$ with $\nabla^{\Sigma}$ being the Levi-Civita connection induced on $\Sigma$ by $g$. How could I approach this? My only sensible calculations, retracing the standard Riccati equation proof, only give me first term and I fail to see the error.