Chapter 12 - Vector-Valued Functions - 12.2 Calculus Of Vector-Valued Functions - Exercises Set 12.2 - Page 857: 54

See proof

\begin{align*} \frac{d}{dt}[u \cdot (v \times w)] &= \frac{d}{dt} [u] \cdot (v \times w) + u \cdot \frac{d}{dt} [(v \times w)] \\ &= \frac{d}{dt} [u] \cdot (v \times w) + u \cdot [v \times \frac{d}{dt} w + \frac{d}{dt} v \times w] \\ &= \frac{du}{dt} \cdot (v \times w) + u \cdot [v \times \frac{dw}{dt}] + u \cdot [\frac{dv}{dt} \times w] \\ \end{align*}