Higher-Order Partials

Calculus 3 · Axiom Academy

LESSON Higher-Order Partial Derivatives Differentiate twice: the pure second partials bend a surface, and the mixed pair f_ xy , f_ yx measure one and the same twist. 1. Notation: Differentiating Twice For a function f(x,y) you can differentiate with respect to x or with respect to y — and then you can differentiate the result again. Two choices, taken twice, give four second-order partial derivatives. Differentiate twice with respect to x . It is the bend of the trace you get by holding y fixed. Differentiate twice with respect to y . The bend of the trace you get by holding x fixed. Differentiate first by x , then by y . A mixed partial: it measures twist, not bend. Differentiate first by y , then by x . The other mixed partial — the subject of Step 2. Twice in the same variable — a pure second partial Once in each variable — a mixed partial 2. Clairaut's Theorem: The Order Doesn't Matter A remarkable fact: for smooth functions, the order of differentiation makes no difference. Differentiate in x then y , or in y then x , and you land on the same function. Why it matters: the theorem cuts your work in half. Instead of computing both mixed partials and hoping they match, you compute whichever one is easier and know the other equals it — provided the second partials are continuous. 3. Verification on a Concrete Function Let's watch Clairaut's theorem happen. Take

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