Estimation D(D_mw)

let $u\in W^{2,2}(\Omega,\mathbb{R}^n)$, $\Omega\subset \mathbb{R}^m$. Does the following inequality (estimation) make sense?

$$\bigg\vert D(\dfrac{\partial u}{\partial x_m})\bigg\vert\leq \vert D^2u\vert$$