Polar Coordinates and Polar Derivatives

In polar coordinates a point is located by a distance and a direction. The point $(r, \theta)$ is $r$ units from the origin, the pole, along the ray at angle $\theta$ from the positive $x$-axis. A negative $r$ means the point is on the opposite ray.

A polar curve $r = f(\theta)$ is the set of points $(f(\theta), \theta)$. As $\theta$ increases, the point sweeps around the pole, and $r$ says how far out it is. Writing $x = f(\theta)\cos\theta$ and $y = f(\theta)\sin\theta$ turns the curve into parametric equations with parameter $\theta$.

The derivative $\frac{dr}{d\theta}$ is the rate at which $r$ changes as $\theta$ increases. Where $r > 0$, a positive $\frac{dr}{d\theta}$ means the point is getting farther from the origin, and a negative one means it's getting closer. Where $r < 0$, the distance is $|r|$, so the signs reverse.

The slope of the tangent line is still $\frac{dy}{dx}$, found from the parametric form with the product rule.

The tangent is horizontal where $\frac{dy}{d\theta} = 0$ and $\frac{dx}{d\theta} \ne 0$, and vertical where $\frac{dx}{d\theta} = 0$ and $\frac{dy}{d\theta} \ne 0$. A horizontal tangent comes from $\frac{dy}{d\theta}$, so $\frac{dr}{d\theta} = 0$ is a different condition. It marks where the distance from the origin stops changing.