$\frac{dy}{dx}= \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \;$ where $\;\frac{dx}{dt} \neq0$

The horizontal tangent occurs when $\;\frac{dy}{dt} =0\;$ given that $\;\frac{dx}{dt} \neq0$.

The vertical tangent occurs when $\;\frac{dx}{dt} =0\;$ given that $\;\frac{dy}{dt} \neq0$.

To find the concavity (or second derivative), we use the following equation:

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