Problem with curvature formula proof. Why this proof doesn't work?

Question

$$\kappa = \frac{||T'(t)||}{||r'(t)||}$$ $$T(t) = \frac{\frac{dr}{dt}}{||\frac{dr}{dt}||}$$ $$T'(t) = \frac{d}{dt} \frac{\frac{dr}{dt}}{||\frac{dr}{dt}||}$$ $$T'(t) = \frac{\frac{d^2r}{dt^2}}{||\frac{dr}{dt}||}$$ $$T'(t) = \frac{r''(t)}{||r'(t)||}$$ $$||T'|| = \frac{||r''(t)||}{||r'(t)||}$$ $$\kappa = \frac{||r''(t)||}{||r'(t)||^2}$$