\begin{frame}
  \frametitle{Derivatives of Logarithmic Functions}
  
  \begin{block}{}
    \begin{malign}
      \frac{d}{dx} (\log_a x) = \frac{1}{x\ln a}
    \end{malign}
  \end{block}
  \pause
  
  \begin{proof}
    \pause
    Let $y = \log_a x$. \pause Then
    \begin{talign}
      a^y = x
    \end{talign}
    \pause
    Using implicit differentiation we get:
    \begin{talign}
      \frac{d}{dx} a^y = \frac{d}{dx} x &\mpause[1]{\quad\implies\quad \ln a \cdot a^y \cdot y'= 1} \\
      &\mpause[2]{\quad\implies\quad y'= \frac{1}{\ln a \cdot a^y} \mpause[3]{= \frac{1}{x\ln a}}} 
    \end{talign}
  \end{proof}
  \pause\pause\pause\pause
  From the formula it follows that
  \begin{block}{}
    \begin{malign}
      \frac{d}{dx} (\ln x) = \mpause[1]{ \frac{1}{x} }
    \end{malign}
  \end{block}
  \vspace{10cm}
\end{frame}