Partial Derivatives

Business Calculus · Axiom Academy

How to differentiate functions of multiple variables When a function depends on multiple variables, we often want to know how the output changes when we change just one input while keeping all others fixed. This is exactly what a partial derivative measures. Definition: Partial Derivative The partial derivative of f(x, y) with respect to x is the derivative of f treating y as a constant. Similarly for the partial derivative with respect to y. Differentiate with respect to x while treating y as a constant There are several common ways to write partial derivatives: How to Compute Partial Derivatives Differentiate with respect to x using standard rules Differentiate with respect to y using standard rules Example 1: Find both partial derivatives of Example 2: Find both partial derivatives of A partial derivative represents the slope of a tangent line to the surface in a specific direction: The slope of the curve you get by slicing the surface with a plane parallel to the xz-plane (holding y constant). The slope of the curve you get by slicing the surface with a plane parallel to the yz-plane (holding x constant). Surface: z = x² + y². The red line shows slope in x-direction, blue line shows slope in y-direction. Marginal Analysis in Multiple Variables If C(x, y) is a cost function depending on two inputs, then: = marginal cost with respect to x (how much cost changes per unit change in x) = marginal cost with respect to y (how much cost changes per unit change in y)

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