Proving that the product rule for differentiating products applies to vectors

[a.merchant]

If r and s are vectors that depend on time, prove that the product rule for differentiating products applies to r.s, that is that:

d/dt (r.s) = r. ds/dt + dr/dt .s


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I'm not entirely sure how I'm supposed to go about proving this, can anyone point me in the right direction, please?

Homework Statement

 

[vela]

Start with

$$\frac{d}{dt} (\vec{r}\cdot\vec{s}) = \frac{d}{dt}(r_x s_x + r_y s_y + r_z s_z)$$