Prove the scalar product rule

mark
Tue, 16 Jan 2018 14:38:24 +0000
Problems

According to theorem 13.2.5(c) in section 13.2 of Guichard's text,
$$\frac{d}{dt} f(t){\bf r}(t)= f(t){\bf r}'(t)+f'(t){\bf r}(t).$$
In this context, $f:\mathbb R\to\mathbb R$ is a scalar function and ${\bf r}:\mathbb R\to\mathbb R^3$ is a vector function.

Write down a componentwise proof of this theorem.