Confusion with unit vectos

Follow the 1st ideia: [tex]\\ d\vec{r} = dr \hat{r} + r d\theta \hat{\theta} \\ \\ \frac{d\vec{r}}{dr} = \frac{dr}{dr} \hat{r} + r \frac{d\theta}{dr}\hat{\theta} \\ \\ \frac{d\vec{r}}{dr} = \hat{r}[/tex] and now: [tex]\\ \frac{d\vec{r}}{dr} = \frac{d\vec{r}}{dt} \frac{dt}{dr} = \frac{d\vec{r}}{dt} \frac{1}{\frac{dr}{dt}} = \frac{\vec{v}}{v} = \hat{v}[/tex] However, is obvious that the unit vectos v and r are different. So, where I'm wrong?