Math Insight

Product rule change in area. Illustration of calculating the derivative of the area $A(t)=x(t)y(t)$ of a rectangle with time varying width $x(t)$ and height $y(t)$. The applet shows the change in area $\Delta A$ over a time interval $\Delta t$ (changeable via a slider), where $\Delta A$ is the area of three rectangles, labeled as $\Delta A_1$ (green rectangle), $\Delta A_2$ (red rectangle), and $\Delta A_3$ (cyan rectangle). As one lets the time interval $\Delta t$ approach zero, the ratio $\Delta A/\Delta t$, approaches the derivative $\diff{A}{t}$. If you change $\Delta t$ to zero, the applet displays the derivative (i.e., the limit as $\Delta t$ approaches zero). As $\Delta t$ shrinks toward zero, the area $A_3$ shrinks to zero much faster than the other two. In fact, even after dividing by $\Delta t$, the ratio $\Delta A_3/\Delta t$ still goes to zero as $\Delta t$ tends to zero. This behavior illustrates the fact that one can ignore $\Delta A_3$ (the cyan rectangle), when calculating the derivative of $A$. Since $\diff{A_1}{t} = \diff{x}{t} y$ and $\diff{A_2}{t} = x\diff{y}{t}$, the applet illustrates the product rule $$\diff{A}{t} = \diff{}{t}(x y) = \diff{x}{t} y + x\diff{y}{t}.$$

More information about applet.