\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}