Questions: Higher-Order Partial Derivatives

Question 1

For f(x, y) = x³y², what operation does the notation ∂²f/∂x∂y instruct you to perform?

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

Answer: (B)

The Leibniz notation ∂²f/∂x∂y is read right-to-left: the variable closest to f (on the right) is differentiated first. So ∂²f/∂x∂y means: first differentiate f with respect to y, then differentiate the result with respect to x. The subscript notation reverses this: f_xy means differentiate x first, then y (left-to-right). So f_xy and ∂²f/∂y∂x represent the same operation. Mixing up these conventions is one of the most common errors with higher-order partials.

Question 2

A student computes f_x = 4x³y for f(x,y) = x⁴y. To find the mixed partial f_xy, she should next differentiate with respect to:

• A. x again, giving 12x²y
• B. y, giving 4x³
• C. both x and y, taking the product of the results
• D. x, then negate the result to account for the mixed direction

Answer: (B)

f_xy means differentiate x first, then y. She has already completed the first step (f_x = 4x³y). The second step is to differentiate with respect to y, treating x as a constant: ∂(4x³y)/∂y = 4x³. This equals f_yx by Clairaut's theorem — verifiable by computing f_y = x⁴, then ∂(x⁴)/∂x = 4x³. ✓

Question 3

If the mixed partials ∂²f/∂x∂y and ∂²f/∂y∂x are both continuous near a point, they must be equal there.

• T. True
• F. False

Answer: True

True — this is Clairaut's theorem. Under the condition that mixed partials are continuous near a point, differentiation order doesn't matter. This is not automatic for all functions — pathological examples exist where the equality fails at points of discontinuity — but for all smooth functions (polynomial, trigonometric, exponential) encountered in practice, mixed partials commute freely.

Question 4

The subscript notation f_xy and the Leibniz notation ∂²f/∂x∂y instruct you to differentiate with respect to x first.

• T. True
• F. False

Answer: False

False — the two notations use opposite reading conventions. In subscript notation f_xy, you differentiate with respect to x first, then y (left-to-right). In Leibniz notation ∂²f/∂x∂y, you differentiate with respect to y first, then x (right-to-left). So f_xy corresponds to ∂²f/∂y∂x in Leibniz notation. Confusing these conventions is one of the most common errors with higher-order partials.

Question 5

Why does the order of differentiation not matter for mixed partials of smooth functions, and when would you need to check whether this equality holds?

Model answer: For smooth functions, Clairaut's theorem guarantees ∂²f/∂x∂y = ∂²f/∂y∂x because the mixed partial measures an interaction — how the rate of change in one direction varies as you move in another — and for well-behaved functions this interaction is symmetric. You need to check whether equality holds when the mixed partials might be discontinuous: at corners, cusps, or points defined piecewise where the function or its derivatives may not be smooth.

The theorem's hypothesis (continuity of mixed partials) is automatically satisfied by all elementary functions everywhere they are defined. Checking becomes necessary only for functions constructed piecewise or near singular points. In practice, verifying continuity of mixed partials is required only in pathological examples designed to violate Clairaut's theorem.