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Choose First Differentiation Variable for Zero 5th-Order Mixed Partial Derivative (4 Functions)
Mathematics
University (Calculus 1/2)
Question Content
The fifth-order partial derivative ∂⁵f/∂x²∂y³ is zero for each of the following functions. To show this as quickly as possible, which variable would you differentiate with respect to first: x or y? Try to answer without writing anything down. a. f(x, y) = y²x⁴eˣ + 2 b. f(x, y) = y² + y(sin x - x⁴) c. f(x, y) = x² + 5xy + sin x + 7eˣ d. f(x, y) = xe^(y²/2)
Correct Answer
a. Differentiate with respect to y first; b. Differentiate with respect to y first; c. Differentiate with respect to y first; d. Differentiate with respect to x first
Detailed Solution Steps
1
Step 1: Analyze each function's polynomial/functional degree in x and y relative to the required partial derivative (2 derivatives with respect to x, 3 with respect to y):
2
Step 2 (Part a): f(x,y) has a maximum degree of 2 in y. Differentiating with respect to y 3 times will immediately eliminate all y-dependent terms, resulting in 0, so prioritize y first.
3
Step 3 (Part b): f(x,y) has a maximum degree of 2 in y. Differentiating with respect to y 3 times will eliminate all y-dependent terms, resulting in 0, so prioritize y first.
4
Step 4 (Part c): All y-dependent terms are degree 1 in y. Differentiating with respect to y 3 times will eliminate these terms, resulting in 0, so prioritize y first.
5
Step 5 (Part d): f(x,y) has a maximum degree of 1 in x. Differentiating with respect to x 2 times will eliminate the x-dependent term, resulting in 0, so prioritize x first.
Knowledge Points Involved
1
Higher-Order Partial Derivatives
Higher-order partial derivatives involve taking partial derivatives of a function multiple times, with respect to the same or different variables. For mixed partial derivatives, Clairaut's theorem states that if the partial derivatives are continuous, the order of differentiation does not matter, but strategically choosing the order can simplify calculations (e.g., eliminating terms early).
2
Polynomial Degree and Differentiation
For a monomial term of the form xⁿyᵐ, differentiating with respect to x k times will eliminate the term if k > n, and differentiating with respect to y m times will eliminate the term if k > m. This property allows quick simplification of partial derivative calculations by choosing the variable with the lower degree first.
3
Partial Derivative Term Elimination
When computing partial derivatives, terms that do not contain the variable being differentiated are treated as constants and their partial derivative is zero. By choosing to differentiate with respect to a variable that appears in only low-degree terms first, we can eliminate those terms entirely in fewer steps, simplifying the rest of the calculation.
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