\begin{frame} \frametitle{Differentiation Rules: Chain Rule} \begin{block}{} \begin{malign} (f\circ g)'(x) \;=\; f'(g(x)) \cdot g'(x) \end{malign} \end{block} \begin{exampleblock}{} Differentiate \begin{talign} f(x) = \left( \frac{x-2}{2x+1} \right)^9 \end{talign} \pause We have \begin{talign} f'(x) &= \mpause[1]{9 \left( \frac{x-2}{2x+1} \right)^8 \frac{d}{dx} \frac{x-2}{2x+1} } \\ &\mpause[2]{ = 9 \left( \frac{x-2}{2x+1} \right)^8 \frac{(2x+1) \cdot 1 - (x-2)\cdot 2}{(2x+1)^2} } \\ &\mpause[3]{ = 9 \left( \frac{x-2}{2x+1} \right)^8 \frac{5}{(2x+1)^2} } \\ &\mpause[4]{ = 45 \frac{(x-2)^8}{(2x+1)^{10}} } \end{talign} \end{exampleblock} \vspace{10cm} \end{frame}