Differentiation of Parametric Functions

Instead of relating $x$ and $y$ directly, it is sometimes more convenient to express both as functions of a third, auxiliary variable — a parameter, usually called $t$. A curve given this way, $x = g(t)$ and $y = f(t)$, is said to be written in parametric form.

For example, $x = at^2$, $y = 2at$ (with $a > 0$ constant and $t \in \mathbb{R}$) parametrize the rightward-opening parabola $y^2 = 4ax$ — substituting confirms every point $(at^2, 2at)$ satisfies that equation. Here $t$ is the parameter; as $t$ varies, the point $(x, y)$ traces out the curve.

Because $x$ and $y$ are each functions of $t$, $y$ effectively becomes a composite function of $x$ through $t$, and its derivative is found using the chain rule: