Parametric Differentiation

Parametric form. Sometimes $x$ and $y$ are both expressed in terms of a third, independent

variable $t$ (the parameter), rather than one being written directly as a function of the

other:

As $t$ varies, the point $(x,y)=(f(t),g(t))$ traces out a curve in the plane. Common examples

include $x=a\cos\theta,\ y=a\sin\theta$ (a circle of radius $a$, parameter $\theta$) and

$x=at^2,\ y=2at$ (a parabola, parameter $t$).

Finding $dy/dx$. Provided $x=f(t)$ and $y=g(t)$ are both differentiable functions of $t$, and

$dx/dt\neq0$, the chain rule (Section 3) gives

by dividing through by $dx/dt$. In words: differentiate $y$ and $x$ separately with respect

to the parameter $t$, then take the ratio -- never differentiate $y$ with respect to $x$

directly when only the parametric equations are given, and never attempt to eliminate $t$ and

find an explicit $y=h(x)$ unless the elimination is genuinely straightforward (it frequently is

not).

Worked illustration ($x=at^2,\ y=2at$, a parabola).

so

matching Example 8 -- the slope of the parabola $y^2=4ax$ at the point corresponding to parameter

$t$.

A second illustration ($x=a\cos\theta,\ y=b\sin\theta$, an ellipse).

so

Why this matters beyond convenience. Many curves that are natural to describe with a

parameter -- circles, ellipses, cycloids, projectile trajectories with $t$ as time -- either have …