\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}