Derivative of Vector Cross Product of Vector-Valued Functions/Proof 1

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: