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 convert that rate into a rate in x — to see why stopping one step early produces a clean and wrong answer, to rule out the ratio of second derivatives, 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, not against x. Converting it takes one more division by dx/dt, and it is the division students leave out, because stopping early produces an expression that simplifies just as cleanly as the right one. Nothing in the algebra objects. The picture does.

8 STEPS · 6 QUICK CHECKS · TWO DIVISIONS, ONE PER DERIVATIVE · v1

differentiate the slope · then divide by dx/dt again

Before you start

What you're looking at

The same curve as Topic 9.1, drawn at equal scale on both axes: x = t² + 1 and y = t³ − 3t for t from −2 to 2. Its first derivative dy/dx = (3t² − 3)/(2t) is already known, and this animation differentiates that.

The question

The slope changes as the particle moves. Differentiating it with respect to t says how fast it changes per unit of time. The second derivative asks something else: how fast per unit of x?

Watch for

Three quadratic curves are drawn through the same point with the same tangent line. Only one of them tracks the curve. The other two are the answers you get by stopping one step early and by matching orders — both of which look perfectly respectable on paper.

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