For a set of parametric equations:

\(x=f(t), y=g(t) \)

The derivative can be found by differentiating each equation with respect to \(t\), and then dividing.

\(\dfrac{dy}{dx} = \dfrac{dy}{dt} ÷ \dfrac{dx}{dt} = \dfrac{dy}{\cancel{dt}} × \dfrac{\cancel{dt}}{dx} \)

\(\implies \boxed{\dfrac{dy}{dx} = \dfrac{g'(t)}{f'(t)}} \)

Equations of tangents and normals

Once the gradient function \(\dfrac{dy}{dx} \) is found, the equation of the tangent or normal at any specific point on the curve can be found in the usual way.

Turning points

Turning points can be found in the usual way, by solving \(\dfrac{dy}{dx} = 0 \).

The solution to the equation \(\dfrac{dy}{dx} = 0 \) will be in terms of the parameter (e.g. \(t\)).

Substituting \(t\) in the original parametric equation and solving will provide the \(x\)- and \(y\)-coordinate of the turning point(s).