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

  We compute the derivative of $f(x) = a^x$:
  \pause
  \begin{talign}
    \alert<6->{f'(x)} &= \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
    \mpause[1]{ = \lim_{h \to 0} \frac{a^{x+h} - a^x}{h}} \\
    &\mpause[2]{ = \lim_{h \to 0} \frac{a^x\cdot a^h - a^x}{h}} 
    \mpause[3]{ = \lim_{h \to 0} a^x\frac{a^h - 1}{h}} 
    \mpause[4]{ = \alert<6->{a^x \cdot \lim_{h \to 0} \frac{a^h - 1}{h}} } 
%     \mpause[5]{ = a^x \cdot f'(0) } 
  \end{talign}
  \pause\pause\pause\pause\pause
  \medskip
  
  Note that
  \begin{talign}
    \lim_{h \to 0} \frac{a^h - 1}{h} = f'(0)
  \end{talign}
  \pause
  \begin{block}{}
    For $f(x) = a^x$ we have 
    \begin{talign}
      f'(x) = f'(0) \cdot a^x
    \end{talign}
  \end{block}
  \pause
  Note that slope is proportional to the function itself.
\end{frame}