\begin{frame}
  \frametitle{Derivative as a Function}
  
  \begin{block}{}
    The \emph{derivative of $f$} is a function $f'$ defined by
    \begin{talign}
      f'(x) = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h}
    \end{talign} 
  \end{block}
  \smallskip
  \begin{itemize}
  \pause
    \item The domain of $f'$ is the set $\{x \mid f'(x) \text{ exists}\}$. 
  \pause\smallskip
    \item Geometrically, $f'(x)$ is the slope of the tangent at $(x,f(x))$.
  \end{itemize}
  \pause
  
  \begin{exampleblock}{}
    Let $f(x) = x^3 - x$. Find a formula for $f'(x)$.
    \pause\medskip
    \begin{talign}
      f'(x) &= \lim_{h\to 0} \frac{f(x+h) - f(x)}{h}
      \mpause[1]{= \lim_{h\to 0} \frac{[(x+h)^3 - (x+h)] - [x^3 - x]}{h}} \\
      &\mpause[2]{= \lim_{h\to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x - h - x^3 + x}{h}} \\
      &\mpause[3]{= \lim_{h\to 0} \frac{3x^2h + 3xh^2 + h^3 - h}{h}} 
      \mpause[4]{= \lim_{h\to 0} (3x^2 + 3xh + h^2 - 1)} \\
      &\mpause[5]{= 3x^2 - 1}
    \end{talign}
  \end{exampleblock}
\end{frame}