\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\vspace{-1ex}
%   
  \begin{talign}
    \frac{d}{dx} \underbrace{f}_{\parbox{1.9cm}{\centerline{outer}\centerline{function}}} 
    \underbrace{(\;\;g(x)\;\;)}_{\parbox{1.9cm}{\centerline{evaluated}\centerline{at inner}\centerline{function}}} 
    = 
    \underbrace{f'}_{\parbox{1.9cm}{\centerline{derivative}\centerline{of outer}\centerline{function}}} 
    \underbrace{(\;\;g(x)\;\;)}_{\parbox{1.9cm}{\centerline{evaluated}\centerline{at inner}\centerline{function}}} 
    \cdot
    \underbrace{g'(x)}_{\parbox{1.9cm}{\centerline{derivative}\centerline{of inner}\centerline{function}}} 
  \end{talign} 
  \pause\medskip
  
%   In Leibnitz notation with $y = f(u)$ and $u = g(x)$:
%   \begin{block}{}
%     \begin{malign}
%       \frac{dy}{dx} \;=\; \frac{dy}{du} \cdot \frac{du}{dx}
%     \end{malign}
%   \end{block}
%   \pause\medskip
  
  In words:
  \begin{itemize}
    \item [] The derivative of the composition of $f$ and $g$ 
      is the derivative of $f$ at $g(x)$ times the derivative of $g$ at $x$.
  \end{itemize}
  \vspace{10cm}  
\end{frame}