'(\vec x)$ must be a matrix, and we'll start writing
  it as $Df$ instead of $f'$. We've actually already dealt with
  problems involving derivatives of multiple-variable functions in
  first-semester calculus. The next few problems are very similar
  to related rates or differential problems from first-semester
  calculus, and we'll see how the derivative $f'$ in $dy=f'(x) dx$
  naturally generalizes to a matrix.</p>

  <div class="problem" id="differential-for-volume-of-a-cylinder}">
    <div class="margi