Derivative of Dot Product of Vector-Valued Functions

Theorem
Let:
 * $\mathbf r: x \mapsto \left\langle{r_1 \left({x}\right), r_2 \left({x}\right), \ldots, r_n \left({x}\right)}\right\rangle$
 * $\mathbf q: x \mapsto \left\langle{q_1 \left({x}\right), q_2 \left({x}\right), \ldots, q_n \left({x}\right)}\right\rangle$

be differentiable vector-valued functions.

The derivative of their dot product is given by:


 * $\dfrac \rd {\rd x} \left({\mathbf r \left({x}\right) \cdot \mathbf q \left({x}\right)}\right) = \mathbf r' \left({x}\right) \cdot \mathbf q \left({x}\right) + \mathbf r \left({x}\right) \cdot \mathbf q' \left({x}\right)$

Also see

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