Estimates of the norm of a derivative of a vector vector field composition

Question

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

Answer 1

This statement is false : let $f(x,y)=(x,y)$ and $g(x,y)=(x+y,x+y)$, then $f\circ g=g$ and $D(f\circ g)(x)\neq 0$, so $\|D(f\circ g)(x)\|\neq 0$. But $|\mathrm{Jac}\,g(x)|=0$. So you can't have $\|D(f\circ g)(x)\| \le\|Df(g(x))\||\mathrm{jac}\,g(x)|$.