Dot Product of Vector-Valued Function with its Derivative

Theorem
Let:
 * $\mathbf f \left({x}\right) = \displaystyle \sum_{k \mathop = 1}^n f_k \left({x}\right) \mathbf e_k$

be a differentiable vector-valued function.

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


 * $\mathbf f \left({x}\right) \cdot \dfrac {\d \mathbf f \left({x}\right)} {\d x} = \left\lvert{\mathbf f \left({x}\right)}\right\rvert \dfrac {\d \left\lvert{\mathbf f \left({x}\right)}\right\rvert} {\d x}$

where $\left\lvert{\mathbf f \left({x}\right)}\right\rvert \ne 0$.

Proof
Then:

Hence the result.