Mathfolis

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

x=rcos⁡θ,y=rsin⁡θx = r\cos\theta,\quad y = r\sin\theta
r2=x2+y2,tan⁡θ=yxr^2 = x^2+y^2,\quad \tan\theta = \frac{y}{x}

Slope of a Polar Curve

dydx=dy/dθdx/dθ=(dr/dθ)sin⁡θ+rcos⁡θ(dr/dθ)cos⁡θ−rsin⁡θ\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} = \frac{(dr/d\theta)\sin\theta + r\cos\theta}{(dr/d\theta)\cos\theta - r\sin\theta}
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.

Practice more of this type— AI-generated · infinite problems

Generate Problems →
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.

Practice more of this type— AI-generated · infinite problems

Generate Problems →
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.

Practice more of this type— AI-generated · infinite problems

Generate Problems →
← Motion with VectorsPolar Area →