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\begin{frame}
  \frametitle{Derivatives of Logarithmic Functions}

  \begin{block}{}
    \begin{malign}
      \frac{d}{dx} (\log_a x) = \frac{1}{x\ln a}
      &&
      \frac{d}{dx} (\ln x) = \frac{1}{x}
    \end{malign}
  \end{block}

  \begin{exampleblock}{}
    Differentiate
    \begin{talign}
      f(x) = \ln |x|
    \end{talign}
    \pause
    We have 
    \begin{talign}
      f(x) = \begin{cases}
        \ln x &\text{for $x > 0$}\\
        \ln (-x) &\text{for $x < 0$}
      \end{cases}
    \end{talign}
    \pause
    Thus
    \begin{talign}
      f'(x) = \begin{cases}
        \mpause[1]{\frac{1}{x}} &\text{for $x > 0$}\\
        \mpause[2]{\frac{1}{-x}\cdot (-1) = \frac{1}{x}} &\text{for $x < 0$}
      \end{cases}
    \end{talign}
    \pause\pause\pause
    Hence 
    \begin{talign}
      \frac{d}{dx} \ln|x| = \frac{1}{x} \quad \text{ for all $x\ne 0$}
    \end{talign}
  \end{exampleblock}
  \vspace{10cm}  
\end{frame}