Derivatives of Parametric Curves

Calculus 2 · Axiom Academy

LESSON Derivatives of Parametric Curves How to find slopes, tangents, and concavity when both x and y are driven by a parameter t. Consider a curve where both coordinates depend on a parameter t — think of t as time and the point (x, y) as something tracing a path. As t ticks forward, x changes at rate dx/dt and y changes at rate dy/dt . A single clock t drives both coordinates 2. Deriving the First-Derivative Formula Start from the chain rule applied to y as a function of x , where x in turn depends on t : Now solve for the thing we actually want, dy/dx . As long as we can divide both sides by it: Geometrically this is exactly a rise over a run. Watch the tangent line slide along the curve below: its little triangle has rise = dy/dt stacked on run = dx/dt , and the tilt you see is their ratio. Take the curve . Differentiate each coordinate with respect to t , then form the ratio: So at t = 2 the point is (4, 8) and the slope is . The animation traces this curve as t increases and glues a tangent line to the moving point, with the slope shown live. Notice how the slope grows as the point climbs the steepening branch. 4. Horizontal and Vertical Tangents The slope is a fraction, so the two special cases come from its two pieces going to zero. Consider the looping curve , whose slope is . Numerator = 0 : dy/dt = 0 (with ). Here 3t^2 - 3 = 0 , so — the points (1, -2) and (1, 2) .

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