Topic 9.2 Second Derivatives of Parametric Equations: find d^2y/dx^2 for a parametric curve by differentiating dy/dx with respect to t and dividing by dx/dt, and use it for concavity (BC).

What this topic is asking

The College Board (Topic 9.2, BC only) extends Topic 9.1 to the second derivative $\frac{d^2y}{dx^2}$ of a parametric curve, which measures concavity just as it does for an ordinary function. The catch is that $\frac{dy}{dx}$ is a function of $t$, not of $x$, so you cannot simply differentiate it again with respect to $t$; you must apply the parametric rule a second time.

The formula and why it has two divisions

A worked second derivative

Using concavity at a point

The most common exam use of $\frac{d^2y}{dx^2}$ is to decide whether a parametric curve is concave up or down at a given $t$. The rule is identical to the function case: evaluate $\frac{d^2y}{dx^2}$ at the parameter value and read its sign. A positive value means the curve bends upward (concave up) at that point; a negative value means it bends downward (concave down). Because $\frac{d^2y}{dx^2}$ is expressed in $t$, you substitute the $t$-value, not an $x$-value. This lets you classify concavity, locate possible inflection points (where $\frac{d^2y}{dx^2}$ changes sign), and describe the shape of the traced curve.

Why the naive formula fails

A tempting but wrong shortcut is $\frac{d^2y}{dx^2} = \frac{d^2y/dt^2}{d^2x/dt^2}$. This does not work because the second derivative is not the ratio of second derivatives; differentiation with respect to $x$ always carries the factor $\frac{1}{dx/dt}$. To see the failure, test it on the worked example: $\frac{d^2y}{dt^2} = 12t^2 - 4$ and $\frac{d^2x}{dt^2} = 2$, whose ratio $\frac{12t^2 - 4}{2} = 6t^2 - 2$ is not the correct answer $2$. The reliable method is always to differentiate the first derivative $\frac{dy}{dx}$ (as a function of $t$) and divide by $\frac{dx}{dt}$ once more. Memorizing the structure "differentiate with respect to $t$, then divide by $\frac{dx}{dt}$" makes both the first and second derivatives fall out of the same operation.

A worked sign-of-concavity question