Final answer:
The second-order partial derivatives of the function f(x, y) = sin(3x/y) with respect to x and y require applying the chain rule and product rule. The derivative f₁₁(x, y) is found by first computing the first derivative with respect to x, and then differentiating again with respect to x.
Step-by-step explanation:
Calculation of Second-Order Partial Derivatives
The student has asked to calculate all four second-order partial derivatives of the function f(x,y) = sin(3x/y). To calculate the second-order partial derivatives, we will need to differentiate twice with respect to each variable involved, which are x and y.
Steps for Finding f₁₁(x, y)
First, we find the first derivative of f with respect to x, denoted as f₁(x, y). Using the chain rule, we get: f₁(x, y) = ∂(sin(3x/y))/∂x = cos(3x/y) * (3/y).
Next, we differentiate f₁(x, y) with respect to x again to obtain the second-order partial derivative f₁₁(x, y): f₁₁(x, y) = ∂(cos(3x/y) * (3/y))/∂x = -sin(3x/y) * (3/y)^2 * 3.
Another Three Second-Order Partial Derivatives
Similarly, the partial derivative of f with respect to y (f₂(x, y)), and subsequently the second-order partial derivatives f₁₂(x, y) and f₂₂(x, y), can be found by applying the chain rule and product rule appropriately.
This process will provide a comprehensive understanding of the curvature behavior and the nature of the function f(x, y) in various directions at different points in the domain.