Problem

Find the instantaneous rate of change of the volume of a cube with respect to the length of its edge, x, when x equals 4 inches.

Solution

The volume of a cube with edge length x is

To visualize what is happening, first we'll consider small changes in edge length when x = 4 inches.

The change in volume with respect to a change in the edge length is the difference quotient

for two edge lengths x₁ and x₂.

The units of these quantities are cubic inches (volume) per inch (length), or

The derivative of V(x) with respect to the variable x shows how the volume changes instantaneously when the edge length is changing.

When x = 4 inches,

This is supported by the calculations in the difference quotient table.