\begin{frame} \frametitle{Derivative as a Function} \begin{exampleblock}{} Where is $f(x) = |x|$ differentiable? \pause\medskip For $x > 0$ we have: \begin{itemize} \pause \item $|x| = x$, \pause \item $|x+h| = x+h$ for small enough $h$. \end{itemize} \pause Thus for $x > 0$ \begin{talign} f'(x) = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h} \mpause[1]{= \lim_{h\to 0} \frac{x+h - x}{h}} \mpause[2]{= \lim_{h\to 0} 1} \mpause[3]{= 1} \end{talign} \pause\pause\pause\pause For $x < 0$ we have: \begin{itemize} \pause \item $|x| = -x$, \pause \item $|x+h| = -x-h$ for small enough $h$. \end{itemize} \pause Thus for $x < 0$ \begin{talign} f'(x) = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h} \mpause[1]{= \lim_{h\to 0} \frac{-x-h + x}{h}} \mpause[2]{= \lim_{h\to 0} -1} \mpause[3]{= -1} \end{talign} \end{exampleblock} \vspace{10cm} \end{frame}