Answer
See proof
Work Step by Step
Since \[ \kappa(s) = \left\| \mathbf{r''}(s) \right\| = \left\| \mathbf{T'}(s) \right\| \] Then \[ \mathbf{N}(s) = \frac{\mathbf{T'}(s)}{\left\| \mathbf{T'}(s) \right\|} = \frac{\mathbf{T'}(s)}{\kappa(s)} \] Hence \[ \frac{d\mathbf{T}}{ds} = \kappa(s)\mathbf{N}(s) \]
