Partial Derivatives, Total Derivative, and Marginal Analysis

1) Why partial derivatives in economics?

Many economic relationships involve more than one variable:

• Production: $Q=f(K,L)$
• Utility: $U=U(x_1,x_2)$
• Demand: $Q_d=f(P,Y)$

Partial derivatives measure the marginal effect of one variable holding others constant.

2) Partial derivatives (definition)

For $z=f(x,y)$:

Example: If $z=x^2y+3y$,

3) Second-order partial derivatives

For “well-behaved” functions, mixed partials are equal: