Applet: 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 $$\diff{A}{t} = \diff{}{t}(x y) = \diff{x}{t} y + x\diff{y}{t}.$$
Applet file: product_rule_change_area.ggb
Applet links
This applet is found in the pages
General information about Geogebra Web applets
This applet was created using Geogebra. In most Geogebra applets, you can move objects by dragging them with the mouse. In some, you can enter values with the keyboard. To reset the applet to its original view, click the icon in the upper right hand corner.
You can download the applet onto your own computer so you can use it outside this web page or even modify it to improve it. You simply need to download the above applet file and download the Geogebra program onto your own computer.