Defining and Differentiating Parametric Equations

A normal function y = f(x) forces one x to give exactly one y, so the curve can never double back on itself. Parametric equations break that rule by introducing a third variable, t, often thought of as time. Both x and y are written as separate functions of t: x = x(t), y = y(t). As t runs forward like a clock ticking, the point (x(t), y(t)) traces a path. Because x and y are each just along for the ride, the path can loop, cross itself, or reverse direction, something no single y = f(x) could ever do.

To find the slope of this path, use the chain rule in reverse. Since y depends on t and x depends on t, dy/dt = (dy/dx)(dx/dt) by the chain rule. Solving for dy/dx gives dy/dx = (dy/dt)/(dx/dt), as long as dx/dt is not zero. You compute dy/dt and dx/dt separately by ordinary differentiation with respect to t, then divide. The result is usually still a function of t, telling you the slope at each moment along the path rather than at each x-value.

This slope formula is the foundation for everything else in the unit: finding horizontal and vertical tangents, computing arc length, and later extending the same divide-by-dx/dt idea to vector-valued functions, where position, velocity, and speed all come from differentiating component functions of t. Get comfortable with this one formula and the rest of the unit is mostly dressing it up in new contexts.