Derivatives of Parametric Functions

Sometimes $x$ and $y$ are not related to each other directly at all, but each is given as a separate function of a third 'helper' variable $t$, called a parameter: $x=f(t)$, $y=g(t)$. As $t$ ranges over some interval $[a,b]$, each value of $t$ produces one point $(x,y)$, and together these points trace out a curve — without $y$ ever being written as an explicit function of $x$. The standard illustration is $x=a\cos t$, $y=a\sin t$: squaring and adding gives $x^2+y^2=a^2\cos^2t+a^2\sin^2t=a^2(\cos^2t+\sin^2t)=a^2$, the equation of a circle of radius $a$ centred at the origin, with $t$ ranging over $[0,2\pi]$. Ever …