I want to prove that
$$ \Delta y \left ( \frac{1}{\sqrt{1+ |\nabla y|^2}}-1 \right) + \nabla y \cdot \nabla \frac{1}{\sqrt{1 + |\nabla y|^2}} = \mathcal O ((| \nabla y| + |\nabla^2 y|)^3) $$ as $ | \nabla y| + |\nabla^2 y| \to 0$.
Here, $$ \nabla = \left ( \frac{\partial}{\partial x_1}, \cdots , \frac{\partial}{\partial x_n} \right) $$ $$ \nabla^j = \left ( \frac{\partial^j}{\partial x_1^j}, \cdots , \frac{\partial^j}{\partial x_n^j} \right) \mbox{ for } j \in \mathbb N $$ $$ \Delta = \sum_{j=1}^n \frac{\partial^2}{\partial x_j^2}.$$
Please help me!