Differentiate a curve through its parameter

Try

A point follows x=t² and y=t³. At t=2 its horizontal and vertical rates are 4 and 12. Predict the slope of the traced curve by dividing vertical change per time by horizontal change per time. This slope is not itself the point’s speed.

Predict

If horizontal velocity becomes zero while vertical velocity remains nonzero, predict the tangent direction. If both vanish, does the same conclusion follow? A ratio 0/0 gives no information without examining the curve or a limiting slope.

Experiment

For x=t²,y=t³ at t≠0, the slope is 3t/2. Differentiate that slope with respect to t, then divide by x′=2t to get the second derivative with respect to x.

t	Point	dy/dx	d²y/dx²
1	(1,1)	3/2	3/4
2	(4,8)	3	3/8
−1	(1,−1)	−3/2	−3/4

Plot the points with direction arrows for increasing t. Two parameter values can have the same x coordinate but different y values, so the whole curve is not a single global function of x. Local branches can still have well-defined slopes.

Understand

A parametric curve is a pair of functions x(t),y(t) on a specified interval. If x′(t)≠0, the local slope is dy/dx=y′/x′. The chain rule gives the second derivative as [d/dt(y′/x′)]/x′. It is not y″/x″: that expression compares accelerations rather than the change of geometric slope per horizontal change.

For the circle x=2cos t,y=2sin t, horizontal velocity is −2sin t and vertical velocity 2cos t. Vertical tangents occur at t=0 and π within [0,2π), at points (2,0) and (−2,0), because only the horizontal component vanishes. Horizontal tangents occur where the vertical component is zero while the horizontal component is nonzero. Radian angle conventions are built into these derivative formulas.

If both parameter derivatives vanish, inspect the actual curve. The parameterization x=t³,y=t³ traces the straight line y=x even at the origin. Its geometric slope there is 1 despite the undefined quotient at t=0. The zero parameter velocity is a pause in this description, not necessarily a cusp in the curve. Other examples, such as x=t²,y=t³ at zero, do produce singular geometry, so there is no universal conclusion from 0/0.

Parameter direction is also information. The descriptions (cos t,sin t) and (cos(−t),sin(−t)) trace the same unit circle in opposite directions. At zero their velocities point upward and downward respectively. Reparameterizing can change speed, starting point and traversal count while preserving a geometric image. Do not infer path length or motion from the image alone without the parameter interval.

The coordinate investigation revisits slope as a ratio of signed coordinate changes. Here the limiting coordinate changes come from a shared parameter, and exceptional zero components require the same care with division.

Apply

In practice, compute x′ and y′ before taking their ratio. State excluded parameter values and analyze them separately when asked. For second derivatives, explicitly show the extra division by x′. For orientation, compare a point and its velocity rather than relying on the order in which a drawing was sketched.

Continue

Next integrate the magnitude of the velocity vector to measure traveled length. This depends on the entire parameter interval and counts retracing, unlike the straight-line distance between endpoints.