Derivative of Dot Product of Vector-Valued Functions

Theorem
Let $\mathbf a: \R \to \R^n$ and $\mathbf b: \R \to \R^n$ be differentiable vector-valued functions.

The derivative of their dot product is given by:


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

Also see

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