← MATH 01131 · Calculus II
MATH 01131 · Parametric & polar (often)
Parametric derivatives
Definition
If x=x(t), y=y(t), then dy/dx = (dy/dt)/(dx/dt) when dx/dt≠0. Second derivative: d/dt(dy/dx) divided by dx/dt.
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Tangents, speed √((x′)²+(y′)²), and arc length use the parameter t.
Eliminate t when possible to recognize the curve, but derivatives can stay parametric.
Horizontal tangents: dy/dt=0 with dx/dt≠0; vertical: dx/dt=0 with dy/dt≠0.
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1.For x=t², y=t³−3t, find dy/dx and points with horizontal tangents.