Example 1

Let $f = (f_i)_{1 \leq i \leq n}$ be a function with values in $\mathbf{R}^n$, defined on an interval $I \subset \mathbf{R}$; each real function $f_i$ is none other than the composite function $\mathrm{pr}_i \circ f$, so is differentiable at the point $x_0$ if $f$ is, and, if so, $f'(x_0) = (f'_i(x_0))_{1 \leq i \leq n}$.