I'm having some trouble connecting two different derivations of velocity in polar coordinates: a geometric method and one involving the product rule of derivatives. Both arrive to the same result, however, I am confused how the product rule relates intuitively to this geometric method (shown in the right of the image below). Essentially, velocity in polar coordinates contains two terms:
$$\vec v=dr \hat r+ r\dot{\theta}\hat \theta$$
The first term specifies change in mangitude (in the parallel direction), while the second term relates to change in direction (perpendicular component of velocity). This can be intuitively derived through considering the derivatives of both components when finding velocity, as shown in the right derivation.
However, I am confused with how the product rule arrives to this same result. Simply using the formula $\frac{d}{dt}(r\hat r)=\dot{r}\hat r+r \frac{d}{dt}\hat r$ seems to split up the vector into both the magnitude and direction terms. I'm not sure how to visualize this geometrically, as shown in the right (looking at the product rule using area). Is there a way to geometrically see how the product rule applied to $r \hat r$ results in two separate terms conveying magnitude and direction, respectively? Would there be a way to connect these two different derivations (such as finding the formula for the product rule through understanding the left derivation)?
Thank you so much for the help and I am more than willing to provide extra clarification if needed!
