\begin{frame} \frametitle{Derivatives of Basic Functions} \begin{block}{Sum Rule} If $f$ and $g$ are differentiable, then \begin{talign} \frac{d}{dx}[f(x) + g(x)] \;=\; \frac{d}{dx}f(x) + \frac{d}{dx}g(x) \end{talign} \end{block} \pause \begin{proof} \begin{malign} \frac{d}{dx}[f(x) + g(x)] &\mpause[1]{ = \lim_{h\to 0} \frac{[f(x+h) + g(x+h)] - [f(x) + g(x)]}{h} } \\ &\mpause[2]{ = \lim_{h\to 0} \frac{[f(x+h) - f(x)] + [g(x+h) - g(x)]}{h} } \\ &\mpause[3]{ = \lim_{h\to 0} \left( \frac{f(x+h) - f(x)}{h} + \frac{g(x+h) - g(x)}{h} \right) } \\ &\mpause[4]{ = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h} + \lim_{h\to 0} \frac{g(x+h) - g(x)}{h} } \\ &\mpause[5]{ = \frac{d}{dx}f(x) + \frac{d}{dx}g(x) } \end{malign} \end{proof} \end{frame}