Applications of Partial Derivatives — Marginal Productivity

A firm's output $Q$ is commonly modelled as a function of two inputs — labour $L$ and capital $K$ — written $Q = f(L,K)$, called a production function. Just as an ordinary derivative gives the marginal cost/revenue of Section 1, a partial derivative of the production function gives the marginal productivity of each individual input, holding the other input fixed.

This is exactly the same marginal-analysis idea as Section 1's marginal cost and marginal revenue, extended to a function of two variables: it isolates the effect of changing one input at a time, which is precisely why the partial derivative — not the ordinary derivative — is the right tool, since output genuinely depends on both inputs simultaneously and neither can be varied without a rule for what happens to the other. …