Introduction to Partial Derivatives

In many real-world situations, an output depends on more than one input. In a high-tech assembly plant, total production may depend on technician labor and machine capacity at the same time. This relationship can be represented by a continuous function P(T, M), where T denotes technician labor input, and M denotes machine capacity. When demand increases, but the budget remains fixed, the manager must determine which input will improve production more efficiently.

Partial derivatives provide a way to study the effect of one variable while holding the other variables constant. To examine the effect of technician labor, machine capacity is kept fixed. The partial derivative of production with respect to technician labor measures the instantaneous rate at which production changes as technician labor increases under the current operating conditions. This value is written as\begin{equation*}\jfrac{\partial P}{\partial T}\end{equation*}

It represents the marginal contribution of technician labor. A larger value indicates that adding more technician input would produce a greater immediate increase in total production, assuming machine capacity does not change.

The manager can also examine the effect of increasing machine capacity while keeping technician labor constant. This is measured by the partial derivative of production with respect to machine capacity,

\begin{equation*}\jfrac{\partial P}{\partial M}\end{equation*}

This quantity represents the instantaneous change in production resulting from a small increase in machine capacity at the current level of labor.

By comparing these two partial derivatives, the manager can identify which input provides the greater marginal benefit. If the partial derivative of production with respect to technician labor is larger, technician labor has a stronger immediate effect. If the partial derivative of production with respect to machine capacity is larger, machine capacity is a more efficient investment. In general, a partial derivative measures how a multivariable function changes with respect to one variable while all other variables remain fixed.