Defining Polar Coordinates and Differentiating in Polar Form BC only

Polar coordinates locate a point by a distance and a direction, with $x = r\cos\theta$ and $y = r\sin\theta$; a point has infinitely many polar names, since adding $2\pi$ changes nothing and a negative $r$ reverses along the ray. $\frac{dr}{d\theta}$ is a real rate, measuring how fast the distance from the origin changes, and it is not a slope: on the circle $r = 3$ it is zero everywhere, which would make a closed curve horizontal at every point.

The repair is a reduction rather than a new formula. Substituting $r = f(\theta)$ gives the parametrisation $x = f(\theta)\cos\theta$, $y = f(\theta)\sin\theta$, so 9.1's quotient $\frac{dy/d\theta}{dx/d\theta}$ applies, with the product rule needed in both numerator and denominator. For $r = 2\cos\theta$ this gives $-\cot 2\theta$, which at $\theta = \frac{\pi}{6}$ is about $-0.577$ while $\frac{dr}{d\theta}$ is $-1$. Polar area sweeps circular sectors rather than vertical strips, giving $\frac{1}{2}\int r^{2}\,d\theta$, in which the one-half comes from the sector, the square from the radius entering twice, and the squaring happens inside the integral.