Gradient estimate of an eigenfunction on a compact Riemannian manifold without boundary

Let $e_{\text{l}}(x)$ be an eigenfunction with respect to the Laplace-Beltrami operator ${\mathrm{\Delta }}_M$ on a compact Riemannian manifold $M$ without boundary: ${\mathrm{\Delta }}_M{e}_{\text{l}}={\text{l}}^2{e}_{\text{l}}$. We show the following gradient estimate of $e_{\text{l}}$: for every $\text{l}\ge 1$, there holds $\l|e_\l|\infty/C\leq |\nabla e\l|\infty\leq C{\l}|e\l|_\infty$,where$C$ is a positive constant depending only on $M$.