Derivative of Vector Cross Product of Vector-Valued Functions

Theorem
Let $\mathbf a: \R \to \R^3$ and $\mathbf b: \R \to \R^3$ be differentiable vector-valued functions in Cartesian $3$-space.

The derivative of their vector cross product is given by:


 * $\map {\dfrac \d {\d x} } {\mathbf a \times \mathbf b} = \dfrac {\d \mathbf a} {\d x} \times \mathbf b + \mathbf a \times \dfrac {\d \mathbf b} {\d x}$

Proof
Let:


 * $\mathbf a: x \mapsto \begin {bmatrix} a_1 \\ a_2 \\ a_3 \end {bmatrix}$


 * $\mathbf b: x \mapsto \begin {bmatrix} b_1 \\ b_2 \\ b_3 \end {bmatrix}$

Then:

Also see

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