Let $\underline{u},\underline{v}$ be [[Fréchet Differentiation|differentiable vector functions]]. Then their [[Dot Product in Real n-Space|dot product]] is differentiable and its derivative is given by $\frac{d}{dt} [\underline{u}(t) \cdot \underline{v}(t)]=\underline{u}(t) \cdot \frac{d}{dt} \underline{v}(t) + \underline{v}(t) \cdot \frac{d}{dt} \underline{u}(t)$