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

  \begin{block}{}
    For $f(x) = a^x$ we have 
    \begin{talign}
      f'(x) = f'(0) \cdot a^x
    \end{talign}
  \end{block}
  \pause\medskip
  
  Using the calculator we can estimate that:
  \begin{talign}
    \mpause[1]{\text{for $a = 2$}} && \mpause[1]{f'(0) = \lim_{h \to 0} \frac{2^h - 1}{h} \approx 0.69} \\
    \mpause[2]{\text{for $a = 3$}} && \mpause[2]{f'(0) = \lim_{h \to 0} \frac{3^h - 1}{h} \approx 1.10} 
  \end{talign}
  \pause\pause\pause\medskip
  
  There is a number $a$ between $2$ and $3$ such that $f'(0) = 1$:\pause
  \begin{block}{}
    \begin{malign}
      \text{$e$ is the number such that} \quad \lim_{h\to 0}\frac{e^h - 1}{h} = 1
    \end{malign}
  \end{block}
  \pause
  The function $e^x$ is the only exponential with slope $1$ at $(0,1)$.
\end{frame}