  In this paper, we study qualitative properties of the fractional
$p$-Laplacian. Specifically, we establish a Hopf type lemma for positive weak
super-solutions of the fractional $p-$Laplacian equation with Dirichlet
condition. Moreover, an optimal condition is obtained to ensure
$(-\triangle)_p^s u\in C^1(\mathbb{R}^n)$ for smooth functions $u$.
