\begin{frame} \frametitle{Derivatives of Logarithmic Functions} \begin{block}{} \begin{malign} \frac{d}{dx} (\log_a x) = \frac{1}{x\ln a} \end{malign} \end{block} \pause \begin{proof} \pause Let $y = \log_a x$. \pause Then \begin{talign} a^y = x \end{talign} \pause Using implicit differentiation we get: \begin{talign} \frac{d}{dx} a^y = \frac{d}{dx} x &\mpause[1]{\quad\implies\quad \ln a \cdot a^y \cdot y'= 1} \\ &\mpause[2]{\quad\implies\quad y'= \frac{1}{\ln a \cdot a^y} \mpause[3]{= \frac{1}{x\ln a}}} \end{talign} \end{proof} \pause\pause\pause\pause From the formula it follows that \begin{block}{} \begin{malign} \frac{d}{dx} (\ln x) = \mpause[1]{ \frac{1}{x} } \end{malign} \end{block} \vspace{10cm} \end{frame}