An inequality for the solution of an elliptic PDE

Question

Assume that $\Omega$ is a bounded convex domain, and $u\in H^2(\Omega)$, How to derive the inequality: $$\left\|D^2 u\right\| \leq C\|\Delta u\| \leq C\|A_{i j}\partial_i \partial_j u\|=C\|\nabla \cdot(A \nabla u)-\nabla A \nabla u\| $$ where $u$ is the solution to the elliptic equation $$ \begin{aligned} -\nabla \cdot A \nabla u=f, & \text { in } \Omega, \\ u=0, & \text { on } \partial \Omega. \end{aligned} $$ I think the first inequality is derived from the Miranda-Talenti estimate. I am confused the second one.