Q:

Derivative of polar curve

We all know a derivative of a function in Cartesian coordinates can be represented (or is defined by) by the slope of a line tangent to the graph. I know that concepts of math are often extended beyond the realm of intuitivity, but I was wondering what the derivative of a polar curve represents, a change in the radius over a change in angle. I'm not convinced it's the tangent line to the graph, because the angle is not a linear measurement and doesn't have units. Does anyone have any intuition for this? Thanks.

A:

Well, you seem to have hit the nail on the head! 
In a Cartesian coordinates, it makes sense that $\displaystyle \frac{\Delta y}{\Delta x}$ would help create a line, because a line is compatible with the horizontal and vertical components of the xy grid.
However, in polar coordinates, a tangent line doesn't really tie well with $r$ and $\theta$ like that, especially when it doesn't pass through the origin.
So how do you find the tangent line to a polar curve, well sorry to let you down, but you'll have to parameterize it first and then go from there. Since tangent lines are using Cartesian bases. 

