Math Insight

The idea of the product rule

The product rule for differentiation.

As discussed in the video, the form of the product rule is so simple because one can ignore any combined effects from changing both factors simultaneously. Instead, one calculates the derivative of a product by considering just the cases where one factor is constant and the other is variable, and adding the results to get the final derivative.

The below applet illustrates why this is true, for the case when calculating the derivative of the area $A(t)=x(t)y(t)$ of a rectangle with time varying width $x(t)$ and height $y(t)$.

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 $$\diff{A}{t} = \diff{}{t}(x y) = \diff{x}{t} y + x\diff{y}{t}.$$