\begin{frame} \frametitle{Derivative as a Function} \begin{exampleblock}{} Where is $f(x) = |x|$ differentiable? \pause\medskip For $x = 0$ \begin{talign} f'(0) = \lim_{h\to 0} \frac{f(0+h) - f(0)}{h} \mpause[1]{= \lim_{h\to 0} \frac{|h|}{h}} \end{talign} \pause\pause We need to look at the left and right limits: \begin{talign} \lim_{h\to 0^-} \frac{|h|}{h} &\mpause[1]{\quad\stackrel{\text{since $h < 0$}}{=}\quad \lim_{h\to 0^-} \frac{-h}{h}} \mpause[2]{= \lim_{h\to 0^-} -1} \mpause[3]{= -1} \end{talign} \pause\pause\pause\pause and \begin{talign} \lim_{h\to 0^+} \frac{|h|}{h} &\mpause[1]{\quad\stackrel{\text{since $h > 0$}}{=}\quad \lim_{h\to 0^+} \frac{h}{h}} \mpause[2]{= \lim_{h\to 0^+} 1} \mpause[3]{= 1} \end{talign} \pause\pause\pause\pause The left and right limits are different.\pause\medskip Thus $f'(0)$ does not exist, and $f(x)$ is not differentiable at $0$.\pause\medskip Hence $f$ is differentiable at all numbers in $(-\infty,0) \cup (0,\infty)$. \end{exampleblock} \vspace{10cm} \end{frame}