What is the difference between unit vectors v and r?

[Jhenrique]

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

 

[D H]

You've made this same mistake before, Jhenrique. Just because $\frac{\partial \theta}{\partial r} \equiv 0$ does not mean that $\frac{d \theta}{d r} = 0$.