Derivatives in Polar Form

Calculus 2 · Axiom Academy

LESSON Derivatives in Polar Form Finding the slope of a tangent line to a polar curve by turning it into a parametric curve and letting the chain rule do the work. Given a polar curve , every angle produces a radius r , and that pair lands at a Cartesian point through the standard conversion. Watch the radius swing through one full turn of the cardioid : at each instant the point's horizontal shadow is and its vertical shadow is . θ is the parameter — it drives both x and y For a parametric curve, the slope is the ratio of the two rates . So we differentiate each coordinate with respect to — each needs the product rule , since both r and the trig factor depend on : Stacking the numerator over the denominator gives the key formula. The animation builds it as two separate quantities — the blue numerator ( ) and the red denominator ( ) — because where each one vanishes is the whole story of the next step. 3. Where Tangents Go Flat or Stand Up Now slide a point around the cardioid and keep the tangent line glued to it. The line is computed directly from the formula at each , so its tilt is . Two special things happen, and the formula predicts both: The tangent goes flat when the numerator is zero (and the denominator is not): . The tangent stands straight up when the denominator is zero (and the numerator is not): . If numerator and denominator are 0 at the same , the slope is the indeterminate — a cusp. Take a limit to find the true tangent direction.

This is the written version of the interactive lesson above. See the full Calculus 2 course.