Example 1

Let $f$ be a real function, $\mathbf{g}$ a vector function, both differentiable at a point $x_0$; the function $\mathbf{g}f$ has a derivative equal to $\mathbf{g}'(x_0)f(x_0) + \mathbf{g}(x_0)f'(x_0)$ at $x_0$. In particular, if $\mathbf{a}$ is constant, then $\mathbf{a}f$ has derivative $\mathbf{a}f'(x_0)$. This last remark, in conjunction with example 1 of I, p. 5, proves that if $\mathbf{f} = (f_i)_{1 \leq i \leq n}$ is a vector function with values in $\mathbf{R}^n$, then for $\mathbf{f}$ to be differentiable at the point $x_0$ it is necessary and sufficient that each of the real functions $f_i$ ($1 \leq i \leq n$) be differentiable there: for, if $(\mathbf{e}_i)_{1 \leq i \leq n}$ is the canonical basis of $\mathbf{R}^n$, we can write $\mathbf{f} = \sum_{i=1}^n \mathbf{e}_i f_i$.