A Lie group $G$ be acting on a Riemannian manifold $M$. Is the map $G\times M\to \mathbb{R}$ given by $(g,x)\mapsto \lVert dg_x\rVert$ continuous?

Question

The norm involved is the operator norm $\lVert T\rVert=\sup\{\lvert T(x)\rVert:|x|\leq 1\}$.

Since $g$ is smooth, of course $x\mapsto \lVert dg_x\rVert$ is continuous. But I am having little trouble in proving the whole continuity.