\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}