Let $C$ be a plane curve parametrized by arc length by $\alpha(s)$, $T(s)$ (unit tangent vector) and $N(s)$ (unit normal vector). Prove that $$\frac{d}{ds} N(s)=-\kappa(s)T(s).$$
I know that since $C$ is a curve parametrized by arc length then the tangent vector $\alpha'(s)$ has unit length which equals $T(s) = \frac{\alpha'(s)}{||\alpha'(s)||}$, thus $N(s) = \frac{T'(s)}{||T'(s)||}$.
How can I prove this? .