e chain rule
  generalizes to all dimensions. Just as before, we'll find that
  the first semester calculus rule will generalize to all
  dimensions if we replace $f'$ with the matrix $Df$. Let's recall
  the chain rule from first-semester calculus.

  <div class="theorem" label="The Chain Rule">
    Let $x$ be a real number and $f$ and $g$ be functions of a
    single real variable. Suppose $f$ is differentiable at $g(x)$
    and $g$ is differentiable at $x$. The derivative of $f\circ g$
    at $x$ is $$(f