Does
$\nabla \| \boldsymbol{v} (\boldsymbol{x}) \| = \frac{1}{\| \boldsymbol{v}(\boldsymbol{x}) \|} \frac{1}{2} \left( \nabla \boldsymbol{v}(\boldsymbol{x})+\nabla \boldsymbol{v}^T (\boldsymbol{x}) \right) \cdot \boldsymbol{v}(\boldsymbol{x})$
where $\|.\|$ stands for euclidean norm and $\cdot$ simple contraction product?