\begin{frame}
  \frametitle{Higher Derivatives}
  
  If $f$ is a function,
  the derivative $f'$ is also a function.
  \pause\medskip
  
  Thus we can compute the derivative of the derivative:
  \begin{talign}
    (f')' = f''
  \end{talign}
  \pause
  The function $f''$ is called \emph{second derivative} of $f$.
  \pause\smallskip
  
%   In Leibnitz notation the second derivative of $y = f(x)$ is written as:
%   \begin{talign}
%     \frac{d}{dx} \left(\frac{dy}{dx} \right) \;=\; \frac{d^2y}{dx^2}
%   \end{talign}
  \begin{exampleblock}{}
    Let $f(x) = x^3 - x$. Find $f''(x)$.
    \pause\medskip
    
    We have seen $f'(x) = 3x^2 - 1$. Thus
    \begin{talign}
      f''(x) &= (f')'(x) \mpause[1]{= \lim_{h\to 0} \frac{f'(x+h) - f'(x)}{h}}\\
      &\mpause[2]{= \lim_{h\to 0} \frac{[3(x+h)^2 - 1] - [3x^2 - 1]}{h}}\\
      &\mpause[3]{= \lim_{h\to 0} \frac{3x^2 + 6xh + 3h^2 - 1 - 3x^2 + 1}{h}}\\
      &\mpause[4]{= \lim_{h\to 0} \frac{6xh + 3h^2}{h}}
      \mpause[5]{= \lim_{h\to 0} (6x + 3h)}
      \mpause[6]{= 6x}
    \end{talign}
  \end{exampleblock}
\end{frame}