AP Calculus BC glossary

Second derivative of a parametric curve

Also called: Parametric second derivative

The second derivative of a parametric curve is the derivative of dy over dx taken again with respect to the parameter t, then divided by dx over dt. It is not the second derivative of y over the second derivative of x. Its sign gives the concavity of the curve.

Because the first derivative $\frac{dy}{dx}$ is itself a function of the parameter $t$ , you differentiate it with respect to $t$ and then divide by $\frac{dx}{dt}$ a second time. The curve is concave up on the parameter interval where the result is positive and concave down where it is negative.

\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}

The mistake

Writing the second derivative as the ratio of the two second derivatives, $\frac{\frac{d^2y}{dt^2}}{\frac{d^2x}{dt^2}}$ . That is not correct. The outer derivative is taken with respect to $t$ , and then you divide by $\frac{dx}{dt}$ , never by $\frac{d^2x}{dt^2}$ .

Appears in: Unit 9: Parametric, Polar, and Vector (BC)