Motivating Example of composition and chain rule
Consider the following composition of the parametric curve: (x,y) = (sin 2t, cos t) with the quadric surface z=x^2+(y/3)^2
As the parameter t varies, it traces out the curve in the x,y plane. Above this point is corresponding point on the surface.
The composition of these functions z= sin^2(2t) +cos^2(t)/9 is the height of the point on the surface at time t.