\begin{frame}
  \frametitle{Differentiation Rules: Product Rule}
 
  \begin{talign}
    h'(x) &= \lim_{\Delta x \to 0} \frac{\Delta h}{\Delta x} 
    &&\hspace{1cm}&&
    \Delta h &= \Delta f\cdot g(x) + f(x) \cdot \Delta g + \Delta f \cdot \Delta g
  \end{talign}
  \pause\smallskip
  
  We compute the limit:
  \pause
  \begin{talign}
    h'(x) &= \lim_{\Delta x \to 0} \frac{\Delta h}{\Delta x} 
    \mpause[1]{= \lim_{\Delta x \to 0} \frac{\Delta f\cdot g(x) + f(x) \cdot \Delta g + \Delta f \cdot \Delta g}{\Delta x} } \\
    &\mpause[2]{= \lim_{\Delta x \to 0} \frac{\Delta f\cdot g(x)}{\Delta x} + \lim_{\Delta x \to 0} \frac{f(x) \cdot \Delta g}{\Delta x} + \lim_{\Delta x \to 0} \frac{\Delta f \cdot \Delta g}{\Delta x}} \\
    &\mpause[3]{= g(x) \lim_{\Delta x \to 0} \frac{\Delta f}{\Delta x} + f(x) \lim_{\Delta x \to 0} \frac{\Delta g}{\Delta x} + \lim_{\Delta x \to 0} \left(\frac{\Delta f}{\Delta x}\cdot \Delta g \right)} \\
    &\mpause[4]{= g(x) f'(x) + f(x) g'(x) + \lim_{\Delta x \to 0} \frac{\Delta f}{\Delta x} \cdot \lim_{\Delta x \to 0} \Delta g} \\
    &\mpause[5]{= g(x) f'(x) + f(x) g'(x) + f'(x) \cdot 0} \\[1ex]
    &\mpause[6]{= g(x) f'(x) + f(x) g'(x)}
  \end{talign}
  \vspace{10cm} 
\end{frame}