\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}