Two Divisions, and the Second Goes Missing

You'll learnto build the second derivative of a parametric curve as two operations — differentiate the slope with respect to the parameter, then divide by dx/dt to turn that rate into a rate in x — to see why stopping one step early gives a clean and wrong answer, and to read concavity off a sign that follows the parameter rather than the x-coordinate.

The slope of a parametric curve is itself a function of the parameter, so it has a rate of change too, and that rate is measured against the parameter rather than against x. Converting it takes one more division, and it is the division students leave out. Nothing in the algebra objects. The picture does. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · TWO DIVISIONS, ONE PER DERIVATIVE · v2

differentiate the slope, then divide by dx/dt a second time

Before you start

What you're looking at

The same curve as Topic 9.1, at equal scale: x = t² + 1 and y = t³ − 3t. Its slope dy/dx = (3t² − 3)/(2t) is already known, and this animation differentiates that.

The question

Differentiating the slope says how fast it changes per unit of t. A second derivative asks something else: how fast per unit of x?

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