The Chain Rule: Problem Solving

A metal rod, when heated, elongates due to linear thermal expansion. The length depends on the initial length, the material’s coefficient of thermal expansion, and the change in temperature of the system.

Here, the temperature of the rod changes quadratically with time.

The objective is to find the instantaneous rate of change of the length of a 2-meter metal rod with a known coefficient of thermal expansion at time exactly 10 seconds.

Because length depends on temperature, and temperature depends on time, the chain rule is applied.

To simplify the chain rule calculation, the differentiation is separated into two distinct components.

First, the length is differentiated with respect to temperature. This differentiation gives a constant value: the product of the initial length and the expansion coefficient.

Second, the temperature is differentiated with respect to time using the power rule, giving a linear rate of change.

According to the chain rule, the final rate is the product of these two components. Substituting the known values at 10 seconds gives the required instantaneous rate of change of the length.