I would like to know one estimate of the euclidian norm of a derivative of a vector field composition. For example: Let $f:\mathbb R^n \to \mathbb R^m$ and $g:\mathbb R^p \to \mathbb R^n$ be two differentiable functions. What may I say about $||D(f\circ g)(x)||$? May I say that $\|D(f\circ g)(x)\| \le\|Df(g(x))\||\mathrm{jac}\,g(x)|$?
Thanks in advance