Polar Derivatives
Unit 9 · Parametric, Polar & Vector
Polar Coordinates and Derivatives
In polar coordinates, a point (r, θ) converts to (r cos θ, r sin θ). To find dy/dx for a polar curve r = f(θ), treat x and y as functions of θ and apply the parametric derivative formula.
Polar ↔ Rectangular Conversions
Slope of a Polar Curve
AP Tip: For horizontal tangents in polar: set dy/dθ = 0 (and dx/dθ ≠ 0). For vertical tangents: set dx/dθ = 0 (and dy/dθ ≠ 0).
Type 1
Converting Between Polar and Rectangular
Use x = r cos θ and y = r sin θ to convert points and equations.
Example 1
Convert the polar point (4, π/3) to rectangular coordinates.
Type 2
Slope of a Polar Curve
Apply the polar slope formula after computing dr/dθ, then evaluate at the given θ.
Example 2
Find dy/dx for r = 2 + sin θ at θ = 0.
Example 3
For r = 1 + cos θ, find the slope at θ = π/2.
Type 3
Horizontal Tangents in Polar
Set dy/dθ = 0 (while dx/dθ ≠ 0) to find angles of horizontal tangents.
Example 4
For r = sin θ on [0, π], find all θ where the tangent line is horizontal.