9.2 Second Derivatives of Parametric Equations

For a parametric curve, the second derivative ${\frac{d^2y}{dx^2}}$ measures how the slope ${\frac{dy}{dx}}$ changes with respect to x, not with respect to t. Since ${\frac{dy}{dx}}$ is itself usually a function of t, you compute the second derivative using ${\frac{d^2y}{dx^2}}$=${\frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}}$ provided ${\frac{dx}{dt}\ne0}$. This formula is just the same parametric derivative rule applied again to the first derivative.

You use the second derivative to study concavity for parametric curves. If ${\frac{d^2y}{dx^2}>0}$, the curve is concave up at that point; if ${\frac{d^2y}{dx^2}<0}$, it is concave down. The main thing to remember is that you do not simply differentiate ${\frac{dy}{dx}}$ once and stop. You must still divide by ${\frac{dx}{dt}}$ because the variable of change is x, not t.