Derivative of Dot Product of Vector-Valued Functions

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

be differentiable vector-valued functions.

The derivative of their dot product is given by:


 * $\dfrac{\mathrm d}{\mathrm dx}\left({ \mathbf{r}\left({x}\right) \cdot \mathbf{q}\left({x}\right) }\right) = \mathbf{r}\,'\left({x}\right)\cdot\mathbf{q}\left({x}\right) + \mathbf{r}\left({x}\right)\cdot\mathbf{q}\,'\left({x}\right)$.

Also see

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