\begin{frame}
  \frametitle{Differentiation Rules: Product Rule}

  Assume that $f$ and $g$ are differentiable at $x$, and define 
  \begin{talign}
    h(x) = f(x) \cdot  g(x)
  \end{talign}
  \pause
  We try to find the derivative of $h$ at $x$:
  \begin{talign}
    \begin{aligned}
    h'(x) = \lim_{\Delta x \to 0} \frac{\Delta h}{\Delta x}
    \end{aligned}
    &&\mpause[1]{\text{where}}&&
    \mpause[1]{
    \begin{aligned}
    \Delta h &= h(x+\Delta x) - h(x) \\
    \Delta f &= f(x+\Delta x) - f(x) \\
    \Delta g &= g(x+\Delta x) - g(x)
    \end{aligned}
    }
  \end{talign}
  \pause\pause
  Then
  \begin{talign}
    \Delta h &= \mpause[1]{ f(x+\Delta x)\cdot g(x+\Delta x) - f(x)\cdot g(x) } \\
             &\mpause[2]{= (f(x) + \Delta f) \cdot (g(x) + \Delta g) - f(x)\cdot g(x)}\\
             &\mpause[3]{= \Delta f\cdot g(x) + f(x) \cdot \Delta g + \Delta f \cdot \Delta g}
  \end{talign}
  
\end{frame}