Given a table of values for a function $f(x,y)$, which of the following best describes how to estimate the mixed partial derivative $f_{xy}(3,2)$?

• A. Take the average of $f(3,2)$ and $f(3,2.2)$ from the table.
• B. Compute the difference quotient of $f_x$ with respect to $y$ near $(3,2)$ using the table values.
• C. Differentiate $f(x,y)$ with respect to $x$ twice using the table values.
• D. Multiply the values of $f_x(3,2)$ and $f_x(3,2.2)$ from the table.

Answer: (B)

Verified step by step guidance

1. Step 1: Understand the concept of mixed partial derivatives. The mixed partial derivative f_{xy}(x, y) represents the rate of change of the partial derivative of f with respect to x (denoted as f_x) as y changes. It is computed by first differentiating f with respect to x and then differentiating the result with respect to y.
2. Step 2: Recall the definition of a difference quotient. To estimate a derivative using table values, we use the difference quotient, which approximates the rate of change of a function. For example, the partial derivative f_x can be approximated as (f(x+h, y) - f(x, y)) / h, where h is a small increment in x.
3. Step 3: To estimate f_{xy}(3, 2), first compute f_x at nearby values of y using the table. For example, calculate f_x(3, 2) and f_x(3, 2.2) using the difference quotient formula: f_x(3, y) ≈ (f(3+h, y) - f(3, y)) / h, where h is a small increment in x.
4. Step 4: Once f_x(3, 2) and f_x(3, 2.2) are computed, use the difference quotient again to approximate the rate of change of f_x with respect to y. This is done by calculating (f_x(3, 2.2) - f_x(3, 2)) / Δy, where Δy is the difference between 2.2 and 2.
5. Step 5: The result from Step 4 provides an estimate for the mixed partial derivative f_{xy}(3, 2). This approach aligns with the correct answer: 'Compute the difference quotient of f_x with respect to y near (3, 2) using the table values.'