\begin{frame} \frametitle{Derivatives} \begin{block}{} The \emph{derivative of a function $f$ at a number $a$}, denoted $f'(a)$, is \begin{talign} f'(a) = \lim_{h\to 0} \frac{f(a+h) - f(a)}{h} \end{talign} if the limit exits. \end{block} \pause \begin{exampleblock}{} Find the derivative of $f(x) = x^2 - 8x + 9$ at number $a$. \pause \begin{talign} f'(a) &= \lim_{h\to 0} \frac{f(a+h) - f(a)}{h}\\ &\mpause[1]{= \lim_{h\to 0} \frac{[(a+h)^2 - 8(a+h) + 9] - [a^2 - 8a + 9]}{h}}\\ &\mpause[2]{= \lim_{h\to 0} \frac{a^2+2ah+h^2 - 8a - 8h + 9 - a^2 + 8a - 9}{h}}\\ &\mpause[3]{= \lim_{h\to 0} \frac{2ah + h^2 - 8h}{h}} \mpause[4]{= \lim_{h\to 0} (2a + h - 8)}\\ &\mpause[5]{= 2a-8} \end{talign} \end{exampleblock} \vspace{15cm} \end{frame}