\begin{frame}
  \frametitle{Differentiation Rules: Chain Rule}

  \begin{block}{Chain Rule}
    If $g$ is differentiable at $x$ and $f$ at $g(x)$, then 
    \begin{talign}
      h &= f \circ g &\text{or equivalently } && h(x) &= f(g(x))
    \end{talign}
    is differentiable at $x$ and
    \begin{talign}
      h'(x) \;=\; (f\circ g)'(x) \;=\; f'(g(x)) \cdot g'(x)
    \end{talign}
  \end{block}  
  \pause\bigskip
  
  Intuition with rates of change:
  \begin{itemize}
  \pause
    \item If $g'(x) = N$. Then $g(x)$ changes $N$ times as much as $x$.
  \pause
    \item If $f'(g(x)) = M$. Then $f(x)$ changes $M$ times as much as $g(x)$.\hspace*{-3ex}
  \pause
    \item Thus $(f\circ g)(x) = f(g(x))$ changes $N\cdot M$ times as much as~$x$.\hspace*{-5ex}\ \ 
  \end{itemize}  
  \vspace{10cm}  
\end{frame}