CalcRef

Parametric, Polar & Vectors · Parametric Curves

Parametric Derivative (dy/dx)

dydx=dy/dtdx/dt=y′(t)x′(t)\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{y'(t)}{x'(t)}

The slope of a parametric curve x = x(t), y = y(t) is the ratio of the derivatives with respect to the parameter t.

Conditions. x'(t) ≠ 0.

Worked examples

Find dy/dx for x = t², y = t³ at t = 1.
  1. dx/dt = 2t, dy/dt = 3t²
  2. dy/dx = 3t²/(2t) = 3t/2
  3. At t = 1: dy/dx = 3/2

Answer: 3/2

Find dy/dx for x = cos t, y = sin t.
  1. dx/dt = -sin t, dy/dt = cos t
  2. dy/dx = cos t/(-sin t) = -cot t

Answer: -cot t

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