Example 4

If $E$ is a topological algebra over $\mathbf{R}$ (cf. Introduction), the product $xy$ of two elements of $E$ is a continuous bilinear function of $(x, y)$; if $\mathbf{f}$ and $\mathbf{g}$ have their values in $E$ and are differentiable at the point $x_0$, then the function $x \mapsto \mathbf{f}(x)\mathbf{g}(x)$ has a derivative equal to $\mathbf{f}'(x_0)\mathbf{g}(x_0) + \mathbf{f}(x_0)\mathbf{g}'(x_0)$ at $x_0$. In particular, if $U(x) = (\alpha_{ij}(x))$ and $V(x) = (\beta_{ij}(x))$ are two square matrices of order $n$, differentiable at $x_0$, their product $UV$ has a derivative equal to $U'(x_0)V(x_0) + U(x_0)V'(x_0)$ at $x_0$ (where $U'(x) = (\alpha'_{ij}(x))$ and $V'(x) = (\beta'_{ij}(x))$).