Let $u \in W^{1, \infty}(\mathbb R^n)$, $\rho \in C^\infty_0(\mathbb R^n)$ and $\rho^\varepsilon(x) = \frac{1}{\varepsilon^n} \rho(x/\varepsilon)$. We define $$u^\varepsilon = \rho^\varepsilon \star u.$$ I am trying to show that $\|\nabla u^\varepsilon \|_\infty \le \|\nabla u\|_\infty$ but I don't really see how to do this. My problem is really to find the explicit form of $\nabla u^\varepsilon$. After that, Young Inequality should help to show the desired inquality (not sure about that, just an intuition).
Clearly, the function $x \mapsto \rho^\varepsilon (x - y) u(y)$ is $C^\infty$ a.e. but I don't see how can I bounded $$|\partial(\rho^\varepsilon)(x - y) u(y)|$$ by an integrable function (to use the Leibniz integral rule). Does enyone have some hint ?