Partial Derivatives — Functions of Two Variables

Every function differentiated so far has depended on a single variable. Many genuine business quantities depend on two or more variables at once — a firm's output depends on both labour and capital; a consumer's utility depends on quantities of several goods. For a function of two variables $z = f(x,y)$, a partial derivative measures how $z$ changes when one variable moves while the other is held fixed — exactly the ordinary derivative, applied one variable at a time.

Because every ordinary differentiation rule (power rule, sum rule, product rule) still applies to whichever variable is being differentiated — the other variable is simply carried through every step like any ordinary constant, e.g. differentiating $3x^2y$ with respect to $x$ treats $3y$ as the constant coefficient, giving $6xy$. …