Second Derivatives of Parametric Equations

With x and y both written in terms of t, dy/dx is already a quotient: (dy/dt) divided by (dx/dt). That quotient is a function of t, not of x, so to find its derivative with respect to x you can't just differentiate it in t and stop. You need d/dx, and the chain rule says d/dx of anything equals its d/dt divided by dx/dt. So the second derivative needs that same division applied a second time.

Mechanically: first find dy/dx as a function of t. Then differentiate that whole expression with respect to t, getting d/dt[dy/dx]. Finally divide by dx/dt one more time. The result, d2y/dx2, is still a function of the parameter t, not of x directly — you read off concavity at a given t-value rather than an x-value. This matters because the curve's shape near a point is only partly described by its slope; the second derivative tells you whether the curve bends upward or downward in y as x increases, which is exactly the role concavity plays for ordinary functions.

This topic sits right after finding dy/dx parametrically, and it feeds directly into analyzing motion: if x and y describe position, the same ideas becomes tools for reading acceleration and curvature, and later for sketching parametric and polar curves accurately. Mastering the two-step division here is what makes those later applications reliable.