Derivative of Scalar Triple Product of Vector-Valued Functions

Theorem
Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be differentiable vector-valued functions in Cartesian $3$-space.

The derivative of their scalar triple product is given by:


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

Also see

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