e previous three problems all focused on exactly the same
  concept. The first looked at the concept graphically, showing
  what it means to write $(f\circ \vec r)(t)=f(\vec r(t))$. The
  second reduced the problem to first-semester calculus. The third
  tackled the problem by considering matrix derivatives. In all
  three cases, we wanted to understand the following problem.

  <div class="quote">
    If $z=f(x,y)$ is a function of $x$ and $y$, and both $x$ and
    $y$ are functions of $t$ (i.e., $\vec r(t