TITLE: What is $\dfrac{dr}{d\theta}$?
QUESTION [6 upvotes]: Suppose we have an equation of a polar curve with usual notation $r=f(\theta).$
I am curious about the geometric meaning of $$\dfrac{dr}{d\theta}=f'(\theta).$$ Also I would like to know the relations between $f'(\theta)$ and tangent at $(r, \theta).$
Thank you.

REPLY [5 votes]: A picture speaks louder than words:



EDIT assume that $\Delta x$ and $\Delta \theta$ are very close to $0$.
In addition, this picture intuitively explains why $\tan \gamma=\frac{r}{r'(\theta)}$ in @Narasimham's answer. (Can you see how?)