Second-Order Partial Derivatives

Business Calculus · Axiom Academy

LESSON Second-Order Partial Derivatives Taking partial derivatives of partial derivatives What Are Second-Order Partials? Just as we can take the second derivative of a single-variable function, we can take partial derivatives of partial derivatives. For a function f(x, y), there are four second-order partial derivatives: Differentiate twice with respect to x Differentiate twice with respect to y Clairaut's Theorem (Symmetry of Mixed Partials) If both mixed partial derivatives are continuous, then: This means the order of differentiation doesn't matter for most functions you'll encounter! Find all second-order partial derivatives of ✓ Clairaut's theorem confirmed! Business Application: Concavity Second-order partial derivatives tell us about the curvature of a surface: : Curvature in the x-direction (at fixed y) : Curvature in the y-direction (at fixed x) : How the x-slope changes as y changes (twist) These are crucial for the second derivative test in optimization, which we'll explore in the next module. The key quantity is the discriminant : D determines whether a critical point is a max, min, or saddle point Business Example: Profit Analysis A company's profit depends on price (p) and advertising (a): : Profit is concave down in price (there's an optimal price) : Profit is concave down in advertising (diminishing returns) : Increasing advertising increases the marginal effect of price

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