Owl: A spiral drawn as r(θ) lives naturally in polar language, but tangent slope is a Cartesian idea: dy/dx. To connect them, write x = r cos θ, y = r sin θ, differentiate both with respect to θ, then form (dy/dθ)/(dx/dθ).
The trap is pretending dy/dx = dr/dθ. Radius change is not the same as rise-over-run in the plane. Product rule on r cos θ and r sin θ is mandatory.
We'll compute dx/dθ and dy/dθ, build dy/dx, and hunt horizontal and vertical tangents on a spiral or cardioid.