 multiplied by the derivative of the inside.
  </div>This is exactly the same as the chain rule in
  first-semester calculus. The only difference is that now we have
  vectors above every variable and function, and we replaced the
  one-by-one matrices $f'$ and $g'$ with potentially larger
  matrices $Df$ and $Dg$. If we write everything in vector
  notation, the chain rule in all dimensions is the EXACT same as
  the chain rule in one dimension.

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    <div class="marginpar">
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