Derivative of Square of Vector-Valued Function

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

The derivative of its square is given by:


 * $\map {\dfrac \d {\d x} } {\mathbf a^2} = 2 \mathbf a \cdot \dfrac {\d \mathbf a} {\d x} = 2 a \dfrac {\d a} {\d x}$

where $a = \norm {\mathbf a}$ is the magnitude of $\mathbf a$.