AP Calculus BC glossary

Parametric derivative

For a parametric curve the slope dy over dx is the ratio of dy over dt to dx over dt. The second derivative is not the ratio of second derivatives: you differentiate the first derivative with respect to t and then divide by dx over dt again.

dydx=dy/dtdx/dt,d2ydx2=ddt(dydx)/dxdt\frac{dy}{dx} = \frac{dy/dt}{dx/dt}, \quad \frac{d^{2}y}{dx^{2}} = \frac{d}{dt}\left(\frac{dy}{dx}\right)\Big/\frac{dx}{dt}

A vertical tangent occurs where dx over dt is zero while dy over dt is not, and a horizontal tangent where the reverse holds. If both vanish at once the point needs closer inspection.

The mistake

Computing the second derivative as the ratio of the two second derivatives. That is wrong; the outer derivative must be taken with respect to t and then divided by dx over dt.

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