Differentiability and a mean-value bound for vector-valued maps

Suppose \(G\subset\mathbb R^n\) is open, \(f:G\to\mathbb R^m\) is a function, and \(x_0\in G\).

1. Carefully define what is meant by the derivative \[ f'(x_0):\mathbb R^n\to\mathbb R^m. \]

2. Let \(I\) be the line segment from \(p\) to \(q\) contained in \(G\), and suppose that \(f\) is differentiable at every point of \(I\). Show that for some \(c\in I\), \[ \|f(q)-f(p)\|_2 \leq \|f'(c)\|\,\|q-p\|_2. \]

Hint: let \(w\) be a unit vector such that \[ \|f(q)-f(p)\|_2=(f(q)-f(p))\mathbin{\cdot}w. \]