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

\begin{talign}
h'(x) = \lim_{\Delta x \to 0} \frac{\Delta h}{\Delta x}
&&\text{where}&&
\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}
Then
\begin{talign}
\Delta h &= \Delta f\cdot g(x) + f(x) \cdot \Delta g + \Delta f \cdot \Delta g
\end{talign}\vspace{-2ex}
\pause
\begin{center}
\begin{tikzpicture}[default]
\coordinate (a) at (0,0);
\coordinate (b) at (3,2);
\coordinate (c) at (5,2.7);
\coordinate (u1) at (0,2);
\coordinate (u2) at (3,2.7);
\coordinate (r1) at (3,0);
\coordinate (r2) at (5,2);
\draw (a) rectangle (b);
\draw[fill=cgreen!25] (r1) rectangle (r2);
\draw[fill=cgreen!25] (u1) rectangle (u2);
\draw[fill=cred!25] (b) rectangle (c);

\draw[gray,decorate,decoration={brace,amplitude=5pt,mirror,raise=2pt}] (0,2.7) -- node[left,xshift=-3mm] {$f(x+\Delta x)$} (0,0);
\draw[gray,decorate,decoration={brace,amplitude=5pt,mirror,raise=2pt}] (0,0) -- node[below,yshift=-3mm] {$g(x+\Delta x)$} (5,0);

\draw[gray,decorate,decoration={brace,amplitude=5pt,mirror,raise=2pt}] (5,0) -- node[right,xshift=3mm] {$f(x)$} (5,2);
\draw[gray,decorate,decoration={brace,amplitude=5pt,mirror,raise=2pt}] (5,2) -- node[right,xshift=3mm] {$\Delta f$} (5,2.7);

\draw[gray,decorate,decoration={brace,amplitude=5pt,mirror,raise=2pt}] (3,2.7) -- node[above,yshift=3mm] {$g(x)$} (0,2.7);
\draw[gray,decorate,decoration={brace,amplitude=5pt,mirror,raise=2pt}] (5,2.7) -- node[above,yshift=3mm] {$\Delta g$} (3,2.7);

\node at ($(b)!.5!(c)$) {$\Delta f\Delta g$};
\node at ($(u1)!.5!(u2)$) {$\Delta f \cdot g(x)$};
\node at ($(r1)!.5!(r2)$) {$\Delta g \cdot f(x)$};
\node at ($(a)!.5!(b)$) {$\underbrace{f(x) \cdot g(x)}_{h(x)}$};
\end{tikzpicture}
\end{center}
\vspace{10cm}
\end{frame}