Answer: The partial derivatives exist and are continuous at the point (2, 5), the function is differentiable at that point.
Given: The function f(x, y) = 2x ln(xy - 9) is differentiable at the point (2, 5) if its partial derivatives with respect to x and y exist and are continuous at that point.
Explanation : Let's find the partial derivatives:
∂f/∂x = 2 ln(xy - 9) + 2x * (y / (xy - 9))
∂f/∂y = 2x * (x / (xy - 9))
Now, let's check their values at the point (2, 5):
∂f/∂x(2, 5) = 2 ln(10 - 9) + 2 * 2 * (5 / (10 - 9)) = 2 ln(1) + 20 = 20
∂f/∂y(2, 5) = 2 * 2 * (2 / (10 - 9)) = 8
Since the partial derivatives exist and are continuous at the point (2, 5), the function is differentiable at that point.