Dot Product of Vector-Valued Function with its Derivative

Theorem
Let:
 * $\map {\mathbf f} x = \ds \sum_{k \mathop = 1}^n \map {f_k} x \mathbf e_k$

be a differentiable vector-valued function.

The dot product of $\mathbf f$ with its derivative is given by:


 * $\map {\mathbf f} x \cdot \dfrac {\map {\d \mathbf f} x} {\d x} = \size {\map {\mathbf f} x} \dfrac {\d \size {\map {\mathbf f} x} } {\d x}$

where $\size {\map {\mathbf f} x} \ne 0$.

Proof
Then:

Hence the result.