Final answer:
To show that the function f(x, y) = 6 x ln(xy - 7) is differentiable at (2, 4), we calculate the partial derivatives and evaluate them at that point. Both partial derivatives are continuous, indicating differentiability.
Step-by-step explanation:
Differentiability of f(x, y) = 6 x ln(xy - 7) at (2, 4)
To show that the function is differentiable at the point (2, 4), we need to demonstrate that the partial derivatives exist and are continuous at that point. Let's calculate the partial derivatives first:
∂f/∂x = 6(ln(xy - 7) + x/y)
∂f/∂y = 6x/y
Next, let's evaluate these partial derivatives at (2, 4):
∂f/∂x = 6(ln(2(4) - 7) + 2/4) = 6(ln(1) + 1/2) = 6(0 + 0.5) = 3
∂f/∂y = 6(2/4) = 3
Both partial derivatives are continuous at (2, 4), which means the function is differentiable at that point.