Partial Derivatives

## One Variable at a Time

When a function depends on several variables, the partial derivative asks: how does the output change when I nudge one input, holding all others constant?

$$f(x,y) = x^2 y + 3xy^2 \qquad \frac{\partial f}{\partial x} = 2xy + 3y^2$$

The $\partial$ symbol replaces $d$ to remind you other variables are frozen. Mechanically, differentiate with respect to the chosen variable and treat everything else as a constant.

Economic interpretation: in a production function $Q(K,L)$, the partial $\partial Q / \partial L$ is the marginal product of labour: extra output from one more unit of labour, holding capital fixed.

## Cross-Partials & Optimization

Cross-partials are second-order mixed partials like $\partial^2 f / \partial x \partial y$. By Young's theorem, if mixed partials are continuous, order does not matter: $f_{xy} = f_{yx}$.

Economically, cross-partials show if inputs are complements (marginal product rises with the other input) or substitutes.

The gradient $\nabla f = (f_x, f_y)$ points to steepest ascent. At an optimum, the first-order conditions require:

$$\text{First-order conditions}: \quad f_x = 0, \quad f_y = 0$$

Common pitfall: A partial is a laboratory experiment holding things constant. In reality, hiring more labour usually changes capital too. For total effects, use the total differential.