Questions: Higher-Order Partial Derivatives and Mixed Partials

Question 1

For f(x,y), what does the notation ∂²f/∂y∂x mean — that is, in what order are the differentiations performed?

• A. Differentiate with respect to y first, then with respect to x
• B. Differentiate with respect to x first, then with respect to y
• C. Take the second derivative with respect to both x and y simultaneously
• D. Differentiate twice with respect to x, then twice with respect to y

Answer: (B)

Notation reads right-to-left: ∂²f/∂y∂x means ∂/∂y(∂f/∂x) — first differentiate with respect to x (the rightmost variable), then differentiate the resulting function with respect to y. This is the most common notational confusion with mixed partials. The expression ∂²f/∂x∂y (reversed) differentiates with respect to y first, then x.

Question 2

A student computes ∂²f/∂x∂y and ∂²f/∂y∂x for a function f and gets different answers. A classmate says: 'You must have made an error — those are always equal.' Who is right?

• A. The classmate — Clairaut's theorem guarantees equality for any function that has partial derivatives
• B. The student — mixed partials always differ because the order of differentiation changes the result
• C. The classmate might be wrong — equality is guaranteed only when both mixed partials exist and are continuous near the point
• D. The student — the two expressions measure fundamentally different geometric quantities and need not agree

Answer: (C)

Clairaut's theorem requires that both mixed partials exist and are *continuous* near the point — it does not apply unconditionally. There exist pathological functions (such as f(x,y) = xy(x²−y²)/(x²+y²) at the origin) where both mixed partials exist at a point but disagree, because the continuity condition fails. For smooth functions encountered in most calculus settings, the condition holds and equality is guaranteed, but the classmate's 'always' is false.

Question 3

The mixed partial derivative ∂²f/∂y∂x at a point measures how the rate of change of f in the x-direction varies as you move in the y-direction.

• T. True
• F. False

Answer: True

Think of ∂f/∂x as the 'x-slope function' that varies across the domain. The mixed partial ∂²f/∂y∂x asks how this x-slope changes as y increases. If ∂²f/∂y∂x > 0, moving in the +y direction makes the function steeper in the x-direction. This 'interaction' interpretation is important for the Hessian: the off-diagonal entries capture how the partial slopes interact across variables, which determines whether a critical point is a saddle point or an extremum.

Question 4

Clairaut's theorem guarantees that mixed partial derivatives commute for any function that has partial derivatives at a point — no additional conditions are required.

• T. True
• F. False

Answer: False

The theorem requires continuity of the mixed partials in a neighborhood of the point, not just their existence at the point. The classic counterexample is f(x,y) = xy(x²−y²)/(x²+y²) for (x,y) ≠ (0,0) and f(0,0)=0: both f_xy(0,0) and f_yx(0,0) exist but equal +1 and −1 respectively, violating equality — and indeed the mixed partials are not continuous at the origin. For smooth (C² or better) functions, the continuity condition is automatic and commutativity holds.

Question 5

What does it mean geometrically or conceptually when a mixed partial ∂²f/∂y∂x is large and positive at a point? What does this tell you about the function's behavior near that point?

Model answer: A large positive ∂²f/∂y∂x means that the slope of f in the x-direction increases rapidly as y increases. In other words, moving in the +y direction makes the function much steeper in the x-direction. This indicates strong interaction between the two variables: the effect of changing x on f depends significantly on the current value of y. Functions with large mixed partials have surfaces that twist sharply — a 'saddle-like' coupling between the axes.

This interaction interpretation is why mixed partials appear in the Hessian matrix's off-diagonal entries and why the Hessian determinant (f_xx · f_yy − f_xy²) captures the trade-off between the pure second derivatives and the cross-coupling. When the mixed partial is zero, the two variables act independently near that point (the function is locally separable). When large, changes in one variable amplify sensitivity to the other.