\begin{frame} \frametitle{Derivatives of Logarithmic Functions} \begin{block}{} \begin{malign} \frac{d}{dx} (\log_a x) = \frac{1}{x\ln a} && \frac{d}{dx} (\ln x) = \frac{1}{x} \end{malign} \end{block} \begin{exampleblock}{} Differentiate \begin{talign} f(x) = \ln |x| \end{talign} \pause We have \begin{talign} f(x) = \begin{cases} \ln x &\text{for $x > 0$}\\ \ln (-x) &\text{for $x < 0$} \end{cases} \end{talign} \pause Thus \begin{talign} f'(x) = \begin{cases} \mpause[1]{\frac{1}{x}} &\text{for $x > 0$}\\ \mpause[2]{\frac{1}{-x}\cdot (-1) = \frac{1}{x}} &\text{for $x < 0$} \end{cases} \end{talign} \pause\pause\pause Hence \begin{talign} \frac{d}{dx} \ln|x| = \frac{1}{x} \quad \text{ for all $x\ne 0$} \end{talign} \end{exampleblock} \vspace{10cm} \end{frame}