\begin{frame} \frametitle{Derivative as a Function} \begin{block}{} The \emph{derivative of $f$} is a function $f'$ defined by \begin{talign} f'(x) = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h} \end{talign} \end{block} \smallskip \begin{itemize} \pause \item The domain of $f'$ is the set $\{x \mid f'(x) \text{ exists}\}$. \pause\smallskip \item Geometrically, $f'(x)$ is the slope of the tangent at $(x,f(x))$. \end{itemize} \pause \begin{exampleblock}{} Let $f(x) = x^3 - x$. Find a formula for $f'(x)$. \pause\medskip \begin{talign} f'(x) &= \lim_{h\to 0} \frac{f(x+h) - f(x)}{h} \mpause[1]{= \lim_{h\to 0} \frac{[(x+h)^3 - (x+h)] - [x^3 - x]}{h}} \\ &\mpause[2]{= \lim_{h\to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x - h - x^3 + x}{h}} \\ &\mpause[3]{= \lim_{h\to 0} \frac{3x^2h + 3xh^2 + h^3 - h}{h}} \mpause[4]{= \lim_{h\to 0} (3x^2 + 3xh + h^2 - 1)} \\ &\mpause[5]{= 3x^2 - 1} \end{talign} \end{exampleblock} \end{frame}