\begin{frame} \frametitle{Differentiation Rules: Product Rule} Lets $f$ and $g$ be linear functions: \begin{talign} f(x) = ax + b && g(x) = cx + d \end{talign} \pause \begin{exampleblock}{} What is the derivative of $f\cdot g$? \end{exampleblock}\vspace{-3ex} \pause \begin{talign} \alert<8->{(f\cdot g)'(x)} &= \frac{d}{dx} [f(x)\cdot g(x)]\\ &\mpause[1]{= \frac{d}{dx} [(ax+b)\cdot (cx+d)] }\\ &\mpause[2]{= \frac{d}{dx} [acx^2 + adx + bcx + bd] }\\ &\mpause[3]{= 2acx + ad + bc} \\ &\mpause[4]{= a(cx + d) + c(ax + b)} \\ &\mpause[5]{= \alert{f'(x) \cdot g(x) + g'(x) \cdot f(x)}} \end{talign} \pause\pause\pause\pause\pause\pause We will now see that this also holds for general $f$ and $g$. \end{frame}