Derivatives of Functions in Parametric Forms

Parametric Relations: A New Way to Connect Variables

Sometimes the relationship between $x$ and $y$ is given neither explicitly as $y = f(x)$ nor implicitly as $F(x, y) = 0$. Instead, both $x$ and $y$ are expressed separately in terms of a third variable — the parameter — which links them through their individual dependence on it.

Writing $x = f(t)$ and $y = g(t)$ expresses the relation between $x$ and $y$ in parametric form, with $t$ as the parameter. For example, the circle $x^2 + y^2 = a^2$ can be written parametrically as $x = a\cos\theta$, $y = a\sin\theta$, where $\theta$ is the parameter.

Finding $\frac{dy}{dx}$ in Parametric Form

Since both $x$ and $y$ are functions of $t$, the chain rule gives

Rearranging yields the fundamental formula:

In terms of the functions $x = f(t)$ and $y = g(t)$, this becomes