Derivative of Scalar Triple Product of Vector-Valued Functions

Theorem
Let:
 * $\mathbf f: x \mapsto \left\langle{f_1 \left({x}\right), f_2 \left({x}\right), \ldots, f_n \left({x}\right)}\right\rangle$
 * $\mathbf g: x \mapsto \left\langle{g_1 \left({x}\right), g_2 \left({x}\right), \ldots, g_n \left({x}\right)}\right\rangle$
 * $\mathbf h: x \mapsto \left\langle{h_1 \left({x}\right), h_2 \left({x}\right), \ldots, h_n \left({x}\right)}\right\rangle$

be differentiable vector-valued functions.

The derivative of their scalar triple product is given by:


 * $\dfrac \rd {\rd x} \left({\mathbf f \cdot \left({\mathbf g \times \mathbf h}\right)}\right) = \dfrac {\rd \mathbf f} {\rd x} \cdot \left({\mathbf g \times \mathbf h}\right) + \mathbf f \cdot \left({\dfrac {\rd \mathbf g} {\rd x} \times \mathbf h}\right) + \mathbf f \cdot \left({\mathbf g \times \dfrac {\rd \mathbf h} {\rd x} }\right)$

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