Derivative of Vector Cross Product of Vector-Valued Functions

Theorem
Let:


 * $\mathbf r: t \mapsto \begin{bmatrix} x \\ y \\ z \end{bmatrix}$


 * $\mathbf q: t \mapsto \begin{bmatrix} \chi \\ \gamma \\ \zeta \end{bmatrix}$

be differentiable vector-valued functions, where:


 * $x,y,z,\chi,\gamma,\zeta$

are (images of) differentiable real functions.

The derivative of the cross product of $\mathbf r$ and $\mathbf q$ is given by:


 * $D_t\left({\mathbf r \left({t}\right) \times \mathbf q \left({t}\right)}\right) = \mathbf{r}\,'\left({x}\right) \times \mathbf{q}\left({x}\right) + \mathbf{r}\left({x}\right) \times \mathbf{q}\,'\left({x}\right)$

Also see

 * Derivative of Dot Product of Vector-Valued Functions
 * Derivative of Product of Real Function and Vector-Valued Function